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Here's the problem:

Let A be the set of infinite-length binary strings that have a finite number of 1s. For example, the string 001110101011110000100110000000000... has all 0s after the twelfth 1 and thus is an element of A. Is the set of A countable or uncountable?

A set is countable if it corresponds with a function whose codomain is the set and whose domain is the set of natural numbers. I don't understand how to proceed from here.

mathtase
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    Hint: if you place a decimal point in front of these and regard them as expressing (in binary) real numbers between $0$ and $1$, all of your numbers would be rational (why?). – lulu Feb 21 '22 at 19:36
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    Alternatively, infinite binary strings are in the usual one-to-one correspondence with subsets of $\Bbb N$; which subsets of $\Bbb N$ are given by these strings? – Greg Martin Feb 21 '22 at 19:41

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