The proof is given below:
But I do not understand number(3), the last paragraph, could anyone explain it for me please?
Many thanks!
2026-03-25 06:11:02.1774419062
A difficulty in understanding that every multitape TM is equivalent to a single tape TM.
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You are storing the contents of several tapes on a single tape (and each of these "virtual tapes" are separated by a special symbol, #, so that the single-tape TM can keep tack of which virtual tape it's reading from at the moment). That means that each of these multiple virtual tapes only gets a finite amount of space. But in the multitape version, there is no such limit. So what happens if the multitape TM we are simulating needs more space than what is available on the virtual tape?
In order to solve this problem, the single-tape machine has to be able to expand the room that any of the virtual tapes have available. It does that by shifting everything on the tape one cell over to make room for one more cell each time one of the virtual tapes needs one more cell.