Countable dense subset of Baire (ω^ω)

Question

Prove that $$ D = \{ f \in \mathbb{N}^\mathbb{N} \mid \exists i \forall j \geq i f(j) = 0 \} $$ is a countable dense subset of $\mathbb{N}^\mathbb{N}$ (Baire).

Here is the theorem about the Seperability of Baire space. How can I start? Actually isn’t it’s clear that for all s finite sequence there at least one x eventually goes with 0? how can I show it more formally?

Answer 1

Every basic open set of Baire space is determined by an initial segment.

So every such sets contains (lots of) sequences that are eventually $0$: after the initial segment condition has been satisfied, we're free to choose coordinates..

So the set of eventually $0$ sequences (or eventually $1$ sequences etc) are dense in Baire space (i.e. $\omega^\omega$)