Q: Elements of countable set Set $S$ consisting of all Cauchy sequences $(a_i)$  with $a_i \in \Bbb N,   \forall i$. 
Is this set S is countable? 
How to check countable set having cauchy sequences? 

A: By definition, a sequence is a Cauchy sequence if for each $\epsilon>0$ there exists an $N\in\mathbb N$ so that for any $m,n>N$, $\left|a_m-a_n\right|<\epsilon$. Now choose $\epsilon<1$. Given that your sequences consist only of natural numbers, what does this imply for $a_m-a_n$?

A: It is similar to all the finite subsets of N. Number of finite subsets of N is countable because  every subset is a subset of (1,2,...,n) and number of subsets of this is $2^n$ for any n. Countable union of countable sets is countable. Every sequence is eventually constant....so do the same.
