Show a series in $\ell^{\infty}$ converges to a point in $c_0$

Question

Let $x =(x_1, x_2, ...) \in \ell^{\infty}$

Suppose $\displaystyle\sum_{n=1}^{\infty}x_n e_n$ converges to $x$ with respect to $\| \cdot\|_{\infty}$. Show that $\displaystyle\lim_{n \rightarrow \infty}x_n=0$.

Things I have tried:

Since $\sum_{n=1}^{\infty}x_n e_n$ converges to $x$, there exists some $N$ such that for all $k>N$,

$$\bigg\|x- \sum_{n=1}^k x_n e_n \bigg\|_{\infty} = \bigg\| \sum_{n=k+1}^{\infty} x_n e_n \bigg\|_\infty < \infty.$$

But now I'm confused about the notation. Not sure how to finish this.

Thanks in advance!

Answer 1

Choose $k_0$ such that $\|x- \sum_{n=1}^k x_n e_n \|_{\infty}<\epsilon$ for all $k \geq k_0$. Then $ \| \sum_{n=k_0+1}^{\infty} x_n e_n \| =\sup \{|x_n|: n >k_0\}$ by definition of the norm in $\ell^{\infty}$. Hence, $|x_n| <\epsilon $ for all $n >k_0$.

[The norm in $\ell^{\infty}$: $\|\sum\limits_{k=1}^{\infty}c_ke_k\|=\|(c_1,c_2,\cdots)\|=\sup \{|c_k|: k=1,2,\cdots\}$].