Slashdot Mirror
IBM Flushes Restroom Patent
jdkane writes "As reported in this CNet news article IBM has quietly eliminated a patent it received on a method for determining who gets to use the bathroom next.
I say Kudos to IBM because it is a relief (no pun intended) to see some common sense prevail in the patent news.
A funny quote from the article is "But just because the patent office granted this and other questionable patents doesn't mean the system is broken".;)"
In the article it describes what has been used in low tech meat counters ever since I can remember.
Stupid things kids do.
Holy shit patent.
You might laugh now about this patent, but I work in Operations Research and such data is vitally important for many businesses. The required number and size of the stalls must determinded a priori and well-timed refill of toilet paper is extremely important from a logistics/cost point of view. Makashura and Miller have proven this optimizing problem NP-hard and the IBM patent contains an appromation algorithm with a garantueed maximum error of 1/5 of the original cost which is very good. If the PR deparment of IBM wouldn't be so stupid the would make BIG bucks with this patent.
It seems that they make the same error here which they made with the personal computer: letting things go without enough patent coverage, giving the rise to power competitors like Microsoft or Compaq. Not very clever.
Owner of a Mensa membership card.
The best suggestion I saw on /. was to pay examiners for each application they could reject.
Another one: patents should be backed up by significant and provable documentation of the actual invention process.
Finally, there should be a bounty (payable by the patenter) for anyone able to break a patent by proving prior art.
The alternative is going to be a general constipation of innovation in those countries that allow software patents.
My blog
SUMMARY
A need therefore exists for an apparatus, system, and method for providing reservations for restroom use in places such as on an airplane, a passenger train or boat where safety concerns exist.
Isn't this situation called "I can't wait any longer, I have to go now"??? And the apparatus in question would be to just stand-up and queue for the bathroom use?
What does the lady from the patent office mean?!
"We dont know who issue the patent".....is it so tough to track who is the issuing officer. I am surprised it isn't done.
Anyways the reporter should have asked why such an absurd patent was issued in the first place and probably put her in the spot by giving many examples of such crappy patents being issued. Sheesh, its almost like the patent office wants us to think its a clerical error.....and the best is they get away with it.
vv
apart from the fact that it applies to bathrooms, how is this any different from the age-old "take a number, take a seat" system used in waiting rooms and supermarket delis around the world? I guess the aparatus (claims 17/26) could be said to be new, but harldy a stroke of genius, especially not in 2000AD. I wouldn't be surprised if the Romans used something like this...
Big Blue didn't want to be known as "Big Blue Water"??
<insert groans here>
Scott
But Nyblod said that it's relatively rare for the patent office to re-examine patents it has issued. The office granted 187,882 patents in 2001 but received just 296 requests to re-examine individual patents, she said.
That's 187,586 short if you ask me.
Really, as much as I don't want to be a flaimbait, the "standard examination" or whatever is not quite good enough.
In this day and age, with the amount of educated people at an all time high, and their education at an all time high too, the patent office has to be extra strict with giving patents.
Though everybody at slashdot already knows this, it can't be stressed enoough:
Patents are more a drag than a boost to creativity.
And just a little rant:
For nine years running, IBM has been the leading recipient of patents from the U.S. patent office.
And
We dedicated that patent to the public so that we could continue focusing on our high-quality patent portfolio.
Seem to be in conflict.IBM and patents seem to me to be a issue of quantity not quality.
...when I said I could make the big bucks by patenting the use of wiping with toilet paper. I mean, heck, I figured I'd be doing the patent office a favor. Surely the place would start to smell better after they were introduced to this novel technique. The only question for them would be whether there was enough room left for the paper, what with heads up in the way.
-Rob
me the shit out o... OH, at fscking last! It's my restroom turn. brb guys!.
...)
(Anyway: I should have my sisters take a look and the patent, they may actually LEARN something
(Anyway 2: Thanks god MS didn't patent it. A bug in these code could really KILL people. And I can foresee the script kiddies all going to the bath as they see fit.)
unfinished: (adj.)
The quote about the system not being broken did not come from IBM, it came from a patent attorney.
Personally, I guess the system isn't "broken", but it would be like a wall with cracks running from top to bottom, it's technically still a wall, but it's not something I'd want to lean on.
s/./, Batman!/
In other news, today American Airlines was granted a patent on a new airliner with more restrooms than seats. When asked about the new plane a company official said, "Well, we aren't exactly sure if someone can use more than one restroom at once, but we are keeping our options open. If necessary, for the sake of public safety and security we have plans to eliminate an addition 10% of seats. Either way we'll show IBM who's boss."
sig
How did the IBM system deal with the human element?
When I'm waiting for a washroom on an aircraft, if I *really* need it in a hurry, perhaps becaue I'm sick.. I will ask the people waiting if I can go ahead. Usually they will say yes, if they can handle it.
LIkewise, I will often let a small child or other person who looks really uncomfortable go ahead of me if I can bear it. It's called sharing. I got there first, okay, but I also sat closer to the washroom, perhaps. Or it took me less time to get out of my chair because I have an aisle seat. First come first serve is not always best; it should be more like most needed goes first.
I suppose if we all had some kind of squid detector (ibm patent #4332123) hooked up to our brains, wired into the aircraft central computer (ibm patent # 98344223), then they could actually let us know before we even realize it that we have to go to the washroom. By analyzing how fast an individual bladder or digestive system is working, it could even schedule such things ahead of time. Pre-boarding screening could be done in order to let people konw they cannot fly, because there will not be washroom time available on the current flight.
Employers could finally get some real figures as to how the "restrooms" are used... A nice Web Based interface with, amongst other widjets, a long list of checkboxes for different activities...
Of course, how many people wold actually admit to some of the things on that list is another matter. :)
Ali
Ph33r m3!!!
Critics have charged that too many patents don't pass the smell test [...]
OK - let's think about this: why would IBM care about people using the bathrooms? Well, this process is a direct analogy to CPU resource management. The article mentions estimated wait time, ability to make and cancel resource reservations, and dangers of numerous stalled processes.
I would bet that some engineer was trying to explain the new resource management algorithm, and used the bathroom analogy. They then registered the patent out of humor, or to ensure nobody later claimed their algorithm was prior art as bathroom usage.
10b||~10b -- aah, what a question!
I was thinking this would be some high tech system involving bladder scanners and anal probes. Then I read...
"As envisioned in the patent, the system would be run by a computer that would assign customers a number based on a first-come, first-served basis"
...and realize it's just another "no tickey, no laundry" system...
::sigh::
Time is what keeps everything from happening all at once.
a method for determining who gets to use the bathroom next
Prior art exists, it's called a line.
Somewhere in the heavens... they are waiting.
Has anyone read that patent???? Operating system - preferably UNIX (!!!) or Windows NT (!!!!!). Network protocol - TCP/IP (!!!!...)
Thanks God, thank you, thank you, thank you that you gave some reason to IBM folks. I cannot imagine what would be the mood in the airplane if some funny joker decided to play the best DoS ever...
Phew! I'm glad they didn't dump that on us. It is such a relief it is not going to happen. I would have been peed off if it happened. It was a crappy patent anyway. It's not like people would have been lining up to implement it.
(Sorry, someone had to get all the obvious puns out of the way. Now we can all get on with the serious issue of discussing this story)
If the pattern goes 9am, 10am, 11am, why isn't noon 12am?
I work for IBM and you wouldn't belive the patents we have every year. Its true we have had the most patents every year for the last 9 years, and most patents are legitamately issued but in some rare cases IBM had some strange paents slip through the hole. I'll try and post are most recent patent list.
Looks like someone at IBM has a twisted sense of humor....
I think the key point of this article is the phrase: "We dedicated that patent to the public so that we could continue focusing on our high-quality patent portfolio," Andrews said.
Now if only the everyone else would realize this...
I also plan to file one for wiping now.
the BM in IBM stood for "Business Machines".
Hmm.
As I read through that list, I thought, 'that has to be illegal, or at least immoral'.
I mean, you would have to be a complete and utter pervert to want to moderate slashdot when you're sitting on the can!
#include "two_kinds_of_shit.joke"
--
The office granted 187,882 patents in 2001
That's more than 500 patents every day. And since getting a patent nowadays is as simple as applying, the numbers are probably rising. Damn. 500 a day. There's no way this system will ever get reformed.
Ok, clearly someone is way underage to be submitting stories to slashdot. The patent office handles literally hundreds of thousands of applications a year. I challenge anyone to find that large of a system with an error rate of zero. From internet routing to CNN to squirrels in headlights, with a large enough sample size, you are going to be able to find problems. With this rationalle, everything is broken.
Now, I've got some real objections to the current patent system. I think lots of things are patentable that shouldn't be. However, even if the laws were changed and patents really tightened up, I guarantee you that some weird ones will still slip through.
So, can we please see an end to the /. sport of finding the most ridiculous patents and waving them around as evidence that patents are "broken"? Even if patents were overhauled and the most extreme /. views prevailed, these same examples would still exist. Sometimes, the issue is the bureaucracy and human error, not a "broken" system.
That said, I do think the system is broken; this is just a case of getting the right conclusion from the wrong evidence ("It would be dangerous to jump off a tall building because it's hard to see exactly where you'd land").
Cheers
-b
If I wanted a sig I would have filled in that stupid box.
Instead of having people sit on the crapper for long times (a few minutes, everything's long when YOU need to), just have a time-shitting agreement. All users will have 1/60'th of a second to do thier business. The last 2 people will be in charge of calling the janitor for floor cleanup (ewwww).
I mean, you would have to be a complete and utter pervert to want to moderate slashdot when you're sitting on the can!
All you really need is 802.11 or a long enough network cable... Erm, not that I know anything about using my laptop while on the john. Sure, those AIM conversations can wait until I get back...
---
DRM is like antifreeze, to the MPAA/RIAA it's sweet, to the consumers it's poison.
Well, yesterday or the day before yesterday I had some good flamer kicking me for being so antipatent. He claimed that I was to one-sided and even didn't read the patents. Well, I don't read all patents shown in /. but some I do and with fair detail. Frankly this one was so shocking that I decided to read it. I HIGHLY recomend to do the same as it is better than any jokes here:
The patent includes the possible use of printers and scanners. I may understand printers but scanners?.. What they intend to scan?
They go all over by decribing a whole fullscale network with Windows NT or UNIX and using TCP/IP. Unfortunately they forget about firewalls, IPSec, secure authentication and so on. So one may wonder what might happen to this network and the effect it will have on passengers...
They even talk about databases... To register what?
These ones killed me: "As shown in FIG. 3B, the information contains field corresponding to, for example, passenger identifier by seat assignment, passenger name, whether the passenger is flying first class, price the passenger paid for the ticket, frequent flyer status, time at which the reservation request is received, reservation number assigned, and current position in the queue.
(...)
Alternatively, the central controller may grant reservation according to a set of rules which determines priority based on information stored in the database such as whether the passenger is flying first class, whether the passenger is a frequent flyer of the airline, the price the passenger paid for his ticket, the location of the passenger's seat in the cabin, etc."
Look at the elitism of the sentences. So one goes to the restroom according to the price he paid for the ticket... Everyone else hold up and don't even imagine to do it in place...
Another out of this world claim is the how they warn the passenger. One of the variants is:"or displayed on a screen in front of the cabin where the safety signs are displayed." For some people, mainly children, this sounds good - so everyone knows that he goes to the toilet... Besides it is quite cool to mix warning signs with the fact one needs to go to the toilet.
The tracking system is also something from the other world: "The central controller may employ a tracking system to detect the movement of a passenger and the status of the restroom. For example, when a passenger leaves his seat and arrives at the restroom in fulfillment of his reservation, a sensor may detect the passenger's departure from his seat and/or arrival at the restroom and transmits the signal to the central controller. Likewise, the central controller may receive a signal when a passenger leaves the restroom. The central controller updates the queue based on the information it receives." And we thought that child tracking was evil... Now even adults are tracked to the toilet.
Besides I don't get why the phone is needed here. Considering the beaurocracies of the crew I wonder if we would see some people crying over the phone that they need the toilet fast and for what reason.
...have a bad sense of humor and money to burn. Some of these patents that have come to light lately seem more like practical jokes, or like two guys betting each other over whether or not the patent will be granted. How else can you explain a patent on a procedure like going to the bathroom?
yes, a movement not to tolerate such assinine claims. Good for IBM, for bringing up the rear ! jon@geek.com
"There are 11 kinds of people: those who know binary, those who don't, and those who could not care less!"
I get it it... *eliminated*
... that they FLUSHED the patent ?
From the patent:
Primary Examiner: Lee, Benjamin C.
I think they need to demote examiners that allow such crap turn into patents!
(This sig intentionally left blank)
The patent system is broken, and the fact that they issued 187,882 patents in a single year is itself evidence of that. Patents are essentially like world records; there is no way so many new and nonobvious creations can be produced in a year.
The USPTO's procedure is to give the benefit of the doubt to the would-be patentee, and then let the courts sort it afterwards if its validity is questioned. This is based on their apparent philosophy that to mistakenly NOT grant a patent that has validity is more harmful than granting an invalid patent, so they prefer to err on the side of the patent applicant.
But reality works the other way. It is more harmful to grant a bad patent than to deny a good patent application. Denying a patent does not necessarily mean the product will not be produced. The rejected patentee can still go ahead and create the product, and the rest of the world is also free to create the product, although they would not have the benefit on the monopoly. On the other hand, granting a patent that should not have been granted prevents or hinders everybody else from producing it (and derivative products) even though they could have thought of it on their own. In addition, sometimes the patentee does not even produce any working models of the product, so the effect is that the patent has caused the product not to be produced at all. Caution should be exercised in favor of the rest of the world, not the patent applicant.
There should be a penalty for submitting a patent that gets rejected on the basis of prior art. If the fine is kept by the USPTO, they will have plenty incentive to search properly for prior art, and the applicants will also have added incentive to search for it. If the patent is actually granted and the applicant uses the patent to extract license fees, and prior art is discovered afterwards, the penalty should be based on the license fees which the patentee has extracted.
Sounds brutal, but by making a patent application you are making an extremely strong claim against the rest of the human race -- that none of the other 6 billion people has done what you have done -- and attempting to put a restriction on their behavior, that you better be damn well sure that your creation is so brilliant that nobody else has done it before. One of the main reason why the USPTO has so little time to review patents is the high volume of trivial patent applications. If you aren't just about 100% sure that no prior art exists, get out of the way and keep the system open for those who do have actual legitimate inventions.
To help enforce the nonobviousness aspect of it, when a patent application is submitted there should be a short description, of maybe 50 words or less that summarizes what the supposed invention does, with care taken not to reveal any of the claims or how it is actually done. Then that summary should be published, after which there is a set time period of maybe a month or two during which the public is allowed to submit documentation or even a working product that does the same thing. If somebody else can come up with a solution in a few weeks based on such a short description, it does not meet the novel and nonobvious criteria. If any the submitted documents or products are substantially similar to the patent application, the patent must be rejected. Some things would only need a short phrase like "online auction" or "one click shopping" or "swinging on a swing" for somebody else to come up with a solution in a few days or even hours. ("Substantially similar" can be defined as whether the submission would be likely to be guilty of patent infringement if the patent were actually granted and the submission created afterwards, and there can be a nominal fee for such submissions to discourage frivilous challenge submissions, if the volume is too high.)
---------
There is inferior bacteria on the interior of your posterior.
It said in the article IBM has been granted 3,411 patents this year, and makes 1.4 billion from royalties off existing patents.
I know slashdotters tend to be supportive of IBM because of their Linux funding, but 3400 patents??? I honestly don't think there are that many good, original ideas that can be discovered in one year. It sounds like IBM is abusing the system : I would strongly suspect that much of what they patent is fairly obvious to those in the know or has already occurred to someone else. But instead, IBM has the patents and plans to knee^H^H^H^H anyone who thinks of an idea they happened to have patented and tries to use it.
This is NOT how it was meant to be. Instead of protecting brilliant inventors who come up with nifty gadgets, these patents are basically a big corporation leaching off the future. With this many patents, I'm sure at least one of them could be argued to apply to just about any conceivable, REAL breakthrough that the REAL inventors make in the computer world. When that happens, IBM will send their lawyers to hit you up for money and you'll have to pay unless you are also a big corporation with enough lawyers on the payroll.
The best suggestion I saw on /. was to pay examiners for each application they could reject.
/. comments by them. And I'd rather have a system that awards too many patents, requiring the lowly peon to challenge the big goliath in a court of law to get a patent busted, than have a system that's abused the other way, preventing the lowly peon from getting deserved patents, while their well-financed competition steals their idea and puts it in practice much faster with their larger capital backing. (disclaimer: I am the lowly peon)
/. crowd. We all know that these big businesses and their patents suck. But think of a million-dollar idea, (like a Jump to Conclusions mat..) and think of what you'd have to do to start a company to go in to business to exploit the idea, assuming you don't already own one. Put yourself in the position of the lowly peon with a ticket to glory, instead of a role of angry-slashdotter-reacting-to-abuse-by-large-corpo rations, and think of how grateful you are for each protection afforded to the small patenter, and how loathe you would be to "reform" those protections away... It's eye-opening.
I think someone pointed out last time, too, that almost all patent applications are rejected the first time around. They send it back to you with a list of reasons they rejected it, you fix it, and send it back. Maybe a couple times, until it's eventually accepted. So it wouldn't really make sense to pay them for each time they sent it back...
IANAPTOPE, but I read a lot of
Finally, there should be a bounty (payable by the patenter) for anyone able to break a patent by proving prior art.
Now that's interesting... but again, that would even further skew the patent system from one that tries to put the lowly peon on equal footing with the Established Interests. Example: OmniCorp has a Patent subdivision of their Legal org. The PTO requires an additional $2k (?) bounty deposit to be staked when first patenting something. That's chump change for OmniCorp, but it may be very tricky for John Q. DocBrown to pony up the deposit, in addition to his now-steeper filing fees (thx PTO reorg). So DocBrown is afraid to file his patent, for fear of losing 3-large if his idea turns out to be stupid, instead of just one.
And when do you get your bounty-deposit back? 20 years later, when the patent expires? Now, the PTO could pay the bounty instead, since they kinda screwed up, but I don't think that would help anyone. You might put in place a system where the PE no longer has any reason to approve a valid app, that may be close to prior art, for fear of getting it busted in court and racking up a demerit point.
Another one: patents should be backed up by significant and provable documentation of the actual invention process.
Umm, it's not that way now? You mean including the inspirational moment, or what? Because you have to have everything else well backed up, if you want to assert that you're the actual inventor, if anyone tries ripping you off.
I've got a suggestion for the
Hadn't thought about this, but you are absolutely right. Thanks for one more
;))
item for my list of "stupid things of Swedish bureaucracy". The apartment
renting "system" still wins the price for the most brain death system ever,
though...
I love Sweden, and most Swedish people are very nice, I just can't stand the
unbelievable amount of stupid rules this country has. The (state) liquor stores
you mention is another example(and have in mind that I don't even like
alcoholic drinks, but I still find it stupid, and useless, I have never seen
more drunk people than in the streets of Stockholm).
Not to mention the insane amount of taxes, that are just given away to the %50
of the population that just "can't work" for some reason... *sigh*
What makes me most sad is that is the nicest country I have been in, and most
people are extremely friendly(specially in the north), but in the end I'll have
to move elsewhere because they can't manage the country in minimally sane
way...
(And no, I'm not an incult USAian that doesn't even want to have an ID card,
I'm from Spain, the country with bigger % of morons in the world...)
There are no a single sane country in all the world? *sigh* (All I ask for is:
no software patents, no DCMA or similar stupid laws, good IT infrastructure eg:
I need broadband, cold weather... maybe Canada, but it's too near the USA to
feel safe, I think Antarctica seem the only option
-- A hacker looking for a nice and quiet place to live in this stupid world
Deep Brown
Table-ized A.I.
if (shorts.isSoiled) reservation.Cancel();
Anyone find it ironic that their stock price went up 11% on the same day?
A funny quote from the article is "But just because the patent office granted this and other questionable patents doesn't mean the system is broken".;)"
It is tremendously important to understand that it is NOT the mission of the patent office to issue 100% valid and fully examined patents. Such an effort isn't possible, given the definition of validity. Nor is it even possible to issue 100% decent, not obviously invalid, patents. The cost of such examination would be prohibitive, and neither the government nor inventors would be willing to subsidize it.
A novelty search (which is all the office can do for the sub-thousand dollar fee it receives and examination doesn't review all, or even the best, prior art -- just what can be found with a decent, reasonable review of the search databases. Examiners in some art areas get quite good at it, because they become quickly familiar with a narrow area of art -- and in others (business methods in particular) cannot get good at it because the scope of their examinations is so wildly broad and uncategorized. The office adapts and does ok.
The system isn't broken, it is acting pretty much as designed. Some places it works better than others, but that is the nature of a human process. The question is not whether bad patents exist (they do), but whether bad patents cause more bad, on balance, than good patents cause good. While that is a reasonable question for debate, it is different from observing from the failure of a few individual examples the supposed bankrupcy of a system at large.
This is something like saying that the entirety of a Unix system is bad, because it contains a single design failure or because it contains a single bug. We know it has both -- and yet we use it because it is an excellent environment in which to work and be productive.
(By the way, tha vast majority of spectacularly whacky patents issue, in part, because they are deemed "harmless" or "mostly harmless" by the examiner -- and unworthy of substantive waste of government resources. )
I have worked at 3 IBM sites and been in many of their buildings and this has happened almost without fail. There are also "blessed" bathrooms where this never happens. I can only imagine that those are reserved for upper management.
I'm trying to teach myself to set people on fire with my mind... Is it hot in here?
...why do people comment a pun, intended or otherwise, with no pun intended? Seems to me that by doing so, the pun is then highlighted, drawing attention to it and implying that the pun IS intended. Why not just let it go, and if some get it, great, and if not, so what!?! (btw - didn't bother with the real article - who gives a shit)
Being able to press a button ay my seat, and having a light go on when it's my turn to go is a great idea. I'm all for it.
I don't think this patent was so silly.
You say that as if it were a bad thing.
I bet American was really flushed with pride over that patent...
Computer Algorithm (Score:4, Interesting)
by forsetti on Saturday October 12, @09:19AM (#4436652)
(User #158019 Info)
OK - let's think about this: why would IBM care about people using the bathrooms? Well, this process is a direct analogy to CPU resource management. The article mentions estimated wait time, ability to make and cancel resource reservations, and dangers of numerous stalled processes.
I would bet that some engineer was trying to explain the new resource management algorithm, and used the bathroom analogy. They then registered the patent out of humor, or to ensure nobody later claimed their algorithm was prior art as bathroom usage.
Hopefully IBM hasn't patented a compression algorithm that involves a trough instead of urinals.
This brings to mind the old poem:
I B.M.
U B.M.
we all B.M.
for I.B.M.
Tell me all about it. It probably includes unbreakable 'repeated one-time pad' rot 13 encryption too? And a method of swinging sideways to decrease discomfort while in line? What about the boredom, maybe it has a claim to use a laserpointer (or a blue led) and a cat to pass the time? And... perhaps one-click access to the stall? What about a hyperlink to direct you form a 'batchroom portal' to the actual stalls? Not to forget the separately patented 'bathroom queue ... on a computer'.
Why didn't they try to get the bathroom queyeing technology into an international standard first?
Or claim it's applicable to JPEG?
When swinging on a swing can be patented or when entertaining your house cat with a laser pointer can be patented, it is most certainly broken.
See the URL below...
Gregory Casamento
## Chief Maintainer for GNUstep
In German, "Öl" means oil, which is an interesting notion when you think about it... Both are argually considered necessities, and both can be used as lubricants; oil in machines, beer in a social settings.
---
Open Source Shirts
Patents are supposed to be for IMPLEMENTATIONS of things. Never were they intended for "ideas" or "business methods".
If you didn't have one you could hold in your hand, you couldn't (originally) get a patent.
Then they applied patents to business methods, and then software. (which at it's heart, is a number)
So now, you can patent a number, and not let anybody else print that number.
I know the hard work decent software takes, I'm an independent software developer myself. But to patent an idea (which is about the most you could claim software to be) is against the original concept of a patent.
You want ideas? They're cheap!
Get a 12 pack, go to a family reunion, and ask everybody for their "best ideas". You'll get (under current law) at least 10 patents some thanksgiving for $8.99.
Crazy. Simply crazy.
I have no problem with your religion until you decide it's reason to deprive others of the truth.
...I got my ticket on Priceline... I need to use the bathroom now, please, please, please.... Oops...
I can't believe they put this suggestion in their patent application. It is sick, sick, sick.
Lets arrest the entire patent office staff, they have obviously gone mad and need to be stopped before they grant Time-Warner the patent on breathing.
-=[ Who Is John Galt? ]=-
Proof techniques #1: Proof by Induction.
This technique is used on equations with "_n" in them. Induction
techniques are very popular, even the military used them.
SAMPLE: Proof of induction without proof of induction.
We know it's true for _n equal to 1. Now assume that it's true
for every natural number less than _n. _N is arbitrary, so we can take _n
as large as we want. If _n is sufficiently large, the case of _n+1 is
trivially equivalent, so the only important _n are _n less than _n. We
can take _n = _n (from above), so it's true for _n+1 because it's just
about _n.
QED. (QED translates from the Latin as "So what?")
- this post brought to you by the Automated Last Post Generator...