Subsequence converging

Doom of Doom

Homework Statement

Consider the sequence $$\left\{ x_{n} \right\}$$.

Then $$x_{n}$$ is convergent and $$\lim x_{n}=a$$ if and only if, for every non-trivial convergent subsequence, $$x_{n_{i}}$$, of $$x_{n}$$, $$\lim x_{n_{i}}=a$$.

Homework Equations

The definition of the limit of a series:
$$\lim {x_{n}} = a \Leftrightarrow$$ for every $$\epsilon > 0$$, there exists $$N \in \mathbb{N}$$ such that for every $$n>N$$, $$\left| x_{n} - a \right| < \epsilon$$.

The Attempt at a Solution

Ok, so I easily see how to show that it $$\lim {x_{n}} = a$$, then every convergent subsequence must also converge to $$a$$.
But I'm stuck on how to show the other way.