Is the set of all functions $f:Z_+\rightarrow Z_+$ that are eventually constant countable.
Solution 1:
Alternatively, consider the following:
For a function $f:Z_+\rightarrow Z_+ $ such that $\exists N \in Z_+ : \forall n > N, f(n) = f(N) $
Let $M = min\{ N : \forall n > N, f(n) = f(N) \}$
and $S(f) = \sum_{i=1}^{M} f(i) + M$
Then for any value $k$, there are only finitely many such functions with $S(f) = k$. List all $f$ with $S(f)=1$ then all $f$ with $S(f)=2$ etc.