Let $c_{0}$ be the space of real-valued sequences $\{x_{n}\}$ which converge to zero, equipped with the metric $d(\{x_{n}\}, \{y_{n}\}) = \sup\{|x_{n} – y_{n}|: n \in \mathbb{N}\}$. Let $e_{k}$ denote the sequence in $c_{0}$ which is identically zero, except for the k-th entry which equals 1.
Prove that the metric space $(c_{0}, d)$ is complete.
Proof:
Consider ${b_n}$, a sequence in $c_0$, which is Cauchy. Then $\forall \epsilon > 0$ $\exists N$, such that if $m \ge N$ and $n \ge N$, then $d(b_m, b_n) < \epsilon$.
So the sequence ${b_n}$ converges to some sequence b. Since all $b_n$ and $b_m$, except of finitely many first elements, are identically zero, b will consist of zeros, except for finitely many elements at the beginning.
Thus $b \rightarrow 0$, so $b \in c_0$. Thus $c_0$ is complete.
Is this proof correct and formal enough?