Let $\Omega\subseteq \mathbb{R}^n$ be a open set. We denote with $C(\overline{\Omega})$ the vector space of bounded and uniformly continuous function on $\Omega$. We define an application $$\rVert \cdot \lVert\colon C(\overline{\Omega})\to\mathbb{R}_+\quad \lVert f \lVert=\sup_\Omega|f(x)|.$$ We know that $$\left (C(\overline{\Omega}),\lVert\cdot\rVert \right)$$ is a Banach space.
I must prove that $C(\overline{\Omega})$ is a closed subspace of $L^\infty(\Omega)$.
Solution
Let $f\in \overline{C(\overline{\Omega})}$, then exists a sequence $\{f_n\}\subseteq C(\overline{\Omega})$ such that $$f_n\to f\quad\text{in}\quad L^\infty(\Omega).$$ Since, in particular, $f_n$ are continuous functions we have that $$f_n\to f\quad\text{in}\quad C(\overline{\Omega}).$$
from here, can I conclude that $f\in C(\overline{\Omega})$? Thanks!