TITLE: Incomplete space
QUESTION [3 upvotes]: I must show that $C_0(\mathbb{R})$ space of all continuous real functions $f:\mathbb{R} \longrightarrow \mathbb{R}$ with compact support is not a complete space endowed by norm
$\|f\|=\sup\limits_{t \in \mathbb{R}}|f(t)|$.
Thank for any help.

REPLY [0 votes]: You could also use Baire's theorem to obtain $n\in \mathbb N$ such that every element of $C_0(\mathbb R)$
would have support in $[-n,n]$.
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