Say to you have bitstream 100 bits in length. That means there's about 2^100 different bitstreams. Now, say you can compress these bitstreams a factor of 100 (two powers of ten). That means you have 2^98 different possible compressed bitstreams. But, the compression is lossless, so you must have a 1 to 1 mapping of unique compressed streams to unique uncompressed streams.
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The proof it can't be done is even simpler.
Say to you have bitstream 100 bits in length. That means there's about 2^100 different bitstreams. Now, say you can compress these bitstreams a factor of 100 (two powers of ten). That means you have 2^98 different possible compressed bitstreams. But, the compression is lossless, so you must have a 1 to 1 mapping of unique compressed streams to unique uncompressed streams.
Uhoh :)