Infinite binary sequences countable set

Question

I know that the set of all binary sequences is uncountable, and I'm asked to prove that the set of all binary sequences that are constant from a certain point ($n\in\mathbb{N}$) is countable, meaning the set: $\{\eta:\eta\in\{0,1\}^{\mathbb{N}}\land\exists n\in\mathbb{N}\forall m>n(\eta(m)=\eta(n))\}$ is countable. How does the fact that all binary sequences in this set are constant from a certain point make it countable?

Answer 1

HINT: For each $n$, the set of binary sequences that are constant from the $n$-th term on is finite; you should be able to write down its actual cardinality without much trouble.

Answer 2

Assume, $E_n$ be the collection of binary sequences which are constant after nth stage. $|E_n| < 2^n \ \forall n \in \mathbb{N} $.
and $X$ the collection of all the binary sequences which are constant after some stage. Then, $X \subseteq \cup_{n \in \mathbb{N}} E_n$ which is countable union of finite sets, hence, countable.

Answer 3

If $\eta$ has the binary number $b$ followed a string of all zeroes starting with the $n^{th}$, map $\eta$ to $2b\in\mathbb N$;

if $\eta$ has the binary number $b$ followed by a string of all ones starting with the $n^{th}$,

map $\eta$ to $2b+1\in\mathbb N$.

This is an injective function from to $\{\eta\}$ to $\mathbb N$, so $\{\eta\}$ is countable.