# Quick question - finishing a convergence proof

1. Oct 8, 2009

### Mosis

1. The problem statement, all variables and given/known data
Prove $$c_0$$ is closed in $$l^{\infty}$$.

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

3. 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?

2. Oct 8, 2009

### Office_Shredder

Staff Emeritus
You can also choose n big enough and a sequence yk so that d(xk, yk) < epsilon/2, and |yn| < epsilon/2. Then |xn| < epsilon. But epsilon is arbitrary, so this is the actual definition of converging to zero.