Checking that $C_{0}(X)$ is a vector space

Question

I am trying to prove that $C_0(X)$ is closed subspace of $C_b(X)$ (bounded continuous functions)

Given, $X$ is locally compact. $C_0(X)$ is the space of all continuous functions $f:X \to F$ (field of X) such that for all $\epsilon \ge 0$, $\{x \in X : |f(x)| \ge \epsilon\}$ is compact.

How to show that for sum of two functions of $C_0(X)$ the set $\{x \in X : |f(x)+g(x)| \ge \epsilon\}$ is compact for all $\epsilon$ ?

Answer 1

Note that $$\{|f+g|\geqslant\varepsilon \}\subset\{|f|\geqslant\varepsilon/2 \}\cup \{|g|\geqslant\varepsilon/2 \}.$$ Since the sets $\{|f|\geqslant\varepsilon/2 \}$ and $\{|g|\geqslant\varepsilon/2 \}$ are compact, so is their union. Finally, $\{|f+g|\geqslant\varepsilon \}$ is a closed subset of a compact set.