Is the set of all finite and countably infinite sequences over {0,1} countable?

Question

Is the set of all finite and countably infinite sequences over {0,1} countable?

From my analysis, I think it is countable. I think of this as the set of all strings from a finite alphabet ?={0,1}, hence ??. (is this a good assumption?).

I later show that I can count each string in the following order: 0,1 (length 1), 00, 01, 10, 11 (length 2) and so on.

However, I am very confused from the initial requirement "finite and countably infinite sequences". Does my method account for the countably infinite strings?

Explanation / Answer

Your solution for proving that ?? is countable is correct. However the set ?? of infinite sequences is not countable, since it is in bijection with the set of subsets of N, which is not countable. See this question.

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