Let $M$ be a bounded subset of $C[0,1]$ with supremum metric. Let $$C'=\left\{ F:[0,1] \to\mathbb R: F(x) = \int f(t) \, dt\right\}. $$ Show that $C'$ is totally bounded.
I started to do it with the characterisation of compact metric space but I ended up finding that f is compact and I couldn't find a way to reach F to be compact...