I considered deleting the post, but this seems more cowardly than just admitting I was wrong. But TIL something!
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You mean integers and real numbers between 0 and 1.
All real numbers would start at 0, 0.1, 0.001, 0.0001.... (a 1:1 match with the set between 0 and 1) all the way to 1, 1.1, 1.01.... Etc.
no, there aren't enough integers to map onto the interval (0,1).
probably the most famous proof for this is Cantor's diagonalisation argument. though as it usually shows how the cardinality of the naturals is small than this interval, you'll also need to prove that the cardinality of the integers is the same as that of the naturals too (which is usually seen when you go about constructing the set of integers to begin with)
No, ey mean real numbers and real numbers. Any interval of real numbers will have enough numbers to be equivalent to any other (infinite ones included)