Approximate unit on $C_0(X)$ converges uniformly on compact sets

Question

Let $(e_{\lambda})_{\lambda \in \Lambda} \subseteq C_0(X)$ with $X$ locally compact an (increasing) approximate unit. I assume that for every compact $K\subseteq X$ we have $\left\Vert 1-e_{\lambda}\right\Vert _{K\text{, }\infty}\rightarrow0$ and I think it should not be difficult to prove, but I failed so far.

Answer 1

Since $K$ is compact, you can construct (by Urysohn's Lemma) $f\in C_0(X)$ with $f|_K=1$. For any $k\in K$, $$ |(1-e_\lambda)(k)|=|f(k)-f(k)e_\lambda(k)|\leq\|f-fe_\lambda\|, $$ so $$ \|1-e_\lambda\|_K\leq\|f-fe_\lambda\|. $$