Take the set of all files of 10 bits. There are 1024 possibilities.
You can compress these files to 9 bits. But you could also compress them to 8, 7, 6, 5, 4, 3, 2, 1, or even 0 bits. In total, that gives you: 1024 possibilities!
Of course, the issue is the initial assumption that the files all contain exactly 10 bits. If we had started with the problem of compressing files between 0 and 10 bits to files with at most 9 bits, the proof is trivial.
Take the set of all files of 10 bits. There are 1024 possibilities.
You can compress these files to 9 bits. But you could also compress them to 8, 7, 6, 5, 4, 3, 2, 1, or even 0 bits. In total, that gives you: 1024 possibilities!
Of course, the issue is the initial assumption that the files all contain exactly 10 bits. If we had started with the problem of compressing files between 0 and 10 bits to files with at most 9 bits, the proof is trivial.