Quick question - finishing a convergence proof

[Mosis]

Homework Statement

Prove c_0 is closed in l^{\infty}.

Homework Equations

A set is closed iff it contains all its accumulation points.

The Attempt at a Solution

Let \left\{x_n\right\} be an accumulation point of c_0. Then for all \epsilon > 0, there exists \left\{y_n\right\} in c_0 such that d(x_n, y_n) = sup|x_n - y_n| < \epsilon. Then |x_n - y_n| < \epsilon for all n. Now since \left\{y_n\right\} converges to 0, there exists an N such that |y_n| < \epsilon for all n bigger than N. But then by reverse triangle inequality,
|x_n|\leq |x_n - y_n| + |y_n| < 2\epsilon.

Now I want to conclude that \left\{x_n\right\} converges to 0, but I'm not sure how to say it. It's clear that since \epsilon was arbitrary, \left\{x_n\right\} should converge, but somehow I don't feel this is satisfactory enough for my 3rd year analysis course. Suggestions?

Edit: why does my LaTex look so ugly?

 

[Office_Shredder]

You can also choose n big enough and a sequence y_k so that d(x_k, y_k) < epsilon/2, and |y_n| < epsilon/2. Then |x_n| < epsilon. But epsilon is arbitrary, so this is the actual definition of converging to zero.