Prove the set of sequences $c_0$ which converge to zero in $l_{\infty}$ is closed.

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.

The solution is correct. Just to beef up this post, I'll sketch a slightly different proof: the complement of $l_0$ is open.

If $x\notin l_0$, let $r=\frac12\limsup_{k\to\infty} |x(k)|$. If $\|x-y\|\le r$, then $$\limsup|y(k)| \ge \limsup_{k\to\infty} |x(k)|-r =r$$ hence $y\notin l_0$.

By the way, this is the first time I see notation $l_0$ used for this subspace; all sources I know use $c_0$. I think $l_0$ is prone to confusion.