Functional Analysis – Prove the Set of Sequences $c_0$ Converging to Zero in $l_{\infty}$ is Closed

functional-analysislp-spaces

Prove the set of sequences which converge to zero in $l_{\infty}$ is closed.

Let $x_n(k)\rightarrow x(k)$ as $n\rightarrow\infty$. With $x_n(k)\in c_0$ and $x(k)\in l_{\infty}$.

Let $\varepsilon>0$. Then there exists an $N>0$ such that
$$\parallel x_N-x\parallel_{\infty}:=\sup_{k\in\mathbb{N}}|x_N(k)-x(k)|\leq\varepsilon.$$

Then we have,
\begin{align}
|x(k)| &= |x(k) – x_N(k) + x_N(k)| \\\\
&\leq |x_N(k) – x(k)| + |x_N(k)| \\\\
&\leq \varepsilon + |x_N(k)|\rightarrow \varepsilon\;\; \text{as}\;\; k\rightarrow\infty.
\end{align}

Therefore since $\varepsilon$ was chosen arbitrarily we can conclude that $x(k)\rightarrow0$ and thus that $x(k)\in c_0$

Can someone check my work on this? It seems too slick and painless to be correct.

Best Answer

One more proof: $f:\ell_\infty \to \mathbb R$, $x\mapsto \lim\sup |x_n|$ is continuous so that $c_0= f^{-1}(\lbrace 0\rbrace)$ is closed.

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