Completeness and Cauchy Sequences I came across the following problem on Cauchy Sequences:

Prove that every compact metric space is complete.

Suppose $X$ is a compact metric space. By definition, every sequence in $X$ has a convergent subsequence. We want to show that every Cauchy sequence in $X$ is convergent in $X$. Let $(x_n)$ be an arbitrary sequence in $X$ and $(x_{n_{k}})$ a subsequence that converges to $a$. Since $(x_{n_{k}}) \to a$ we have the following: $$(\forall \epsilon >0) \ \exists N \ni m,n \geq N \implies |x_{n_{m}}-x_{n_{n}}| < \epsilon$$
Using this, we can conclude that every Cauchy sequence in $X$ is convergent in $X$? Or do we inductively create subsequences and use Cauchy's criterion to show that it converges? 

Let $\epsilon > 0$. Since $(x_n)$ is Cauchy, exists $\eta_1\in \mathbb N$ such that
$$ \left\vert x_n - x_m\right\vert < \frac \epsilon 2$$
for each pair $n, m > \eta_1$.
Since $x_{k_n} \to a$, exists $\eta_2 \in \mathbb N$ such that
$$ \left\vert x_{k_n} - a\right\vert < \frac \epsilon 2$$
for each $n > \eta_2$.
Let $\eta = \max\{\eta_1, \eta_2\}$, if $n > \eta$ then $k_n \ge n > \eta$. Therefore we have
$$ \left\vert x_n - a\right\vert \le \left\vert x_n - x_{k_n}\right\vert + \left\vert x_{k_n} - a\right\vert < \frac \epsilon 2 + \frac \epsilon 2 = \epsilon$$
