Let $C_b([0,\infty))$ be the space of bounded continuous functions on $[0,\infty)$. On $C_b([0,\infty))$ we define a norm by $\|f\|_\infty = \sup\{|f(x)| : x\in [0,\infty)\}$. Furthermore, define \begin{equation} C_0([0,\infty)) = \{f \in C_b([0,\infty)):\lim\limits_{n \to \infty}f(x) = 0\} \end{equation}
Exercise: Show that $C_0([0,\infty)), \|\cdot\|_\infty)$ is complete.
Hint: you can use that $(C_b([0,\infty)),\|\cdot\|_\infty)$ is a complete space.
I know that:
If $(C_b([0,\infty)),\|\cdot\|_\infty)$ is a complete space, $C_0([0,\infty)), \|\cdot\|_\infty)$ is complete if $C_0([0,\infty))$ is closed in $C_b([0,\infty))$.
$C_0([0,\infty))$ is closed $\Leftrightarrow$ $B_\epsilon(x) \cap C_0([0,\infty)) \neq \emptyset$ for any $\epsilon > 0$, then $x\in C_0([0,\infty))$.
What I think I should do:
Pick $x\in C_b([0,\infty))$ such that $B_\epsilon(x)\cap C_0([0,\infty)) \neq \emptyset$ for every $\epsilon >0$. Then; show that $x\in C_0([0,\infty))$ as well.
Question: How do I show that $(C_0([0,\infty)), \|\cdot\|_\infty)$ is complete by using my approach and showing that $x\in C_0([0,\infty))$ as well?