Prove Uncountable

Question

For this question you are not allowed to invoke any set that is known to be uncountable (such as subsets of $\mathbb{R}$) in your answer. Let $A = \{a, b, c\}$. Consider the set of functions $F$ whose elements are functions $f : \mathbb{Z}^+ → A$. Using diagonalization, prove that $F$ is uncountable.

For this I think I need to do cantors diaganolization but I'm not sure what "$f : \mathbb{Z}^+ → A$" means and how to start it off

Thanks

Answer 1

Functions $f:\mathbb{Z}^+\to A$ are functions whose inputs are positive integers, and whose outputs are the elements of $A$.

In other words: they are sequences with elements in $A$.

The point here is to almost verbatim repeat the classic diagonalization argument, except that your sequence has 3 different values ($a,b,c$) rather than 2 ($0,1$).