Show that $C_0([a, b], \mathbb{R})$ is not $\sigma$-compact

$C_0([a, b], \mathbb{R})$ is the space of real-valued continuous function on $[a, b]$. The hint says think Baire. So I assume that $C_0([a, b], \mathbb R)$ is $\sigma$-compact. Then it is the countable union of compact, and hence closed, subsets $M_i$. Since $C_0([a, b], \mathbb R)$ is a complete metric space, by Baire's theorem at least one of the $M_i$'s has a non-empty interior. But I got stuck here. I think I can derive a contradiction from this, but I'm not sure how. Any hints on how to solve this (using contradiction or not) would be much appreciated.

Solution 1:

Hint: Show that the closed unit ball is not compact. Now conclude that if $U$ is a compact set then it has an empty interior.