how to determine the following set is countable or not? How to determine whether or not these two sets are countable?

The set A of all functions $f$ from $\mathbb{Z}_{+}$ to $\mathbb{Z}_{+}$.
The set B of all functions $f$ from $\mathbb{Z}_{+}$ to $\mathbb{Z}_{+}$ that are eventually 1.

First one is easier to determine since set of fuctions from $\mathbb{N}$ to $\{0,1\}$ is uncountable. But how to determine the second one? Thanks in advance!

Let $B_n$ be the set of functoins $f\colon \mathbb Z_+\to\mathbb Z_+$ with $f(x)\le n$ for all $x$ and $f(x)=1$ for $x>n$.
Then $$B=\bigcup_{n\in\mathbb N}B_n$$
and $|B_n|=n^n$.
