Let $E=C[a.b]$ with the max norm $$\|f\|_0 = \max_{x\in[a,b]} |f(x)|$$ and the operator $T:(E,\|\cdot\|_0) \rightarrow (E,\|\cdot\|_0)$ $$T(f)(x)=\int_a^x f(s) \, ds, \ \forall \ x \in [a,b].$$ For $M>0$, let $F \subset E$ $$F = \{T(f); f \in E \ \text{e} \ \|f\|_0 \leq M\}.$$ Let $f_n$ be a sequence in $F$, show that exists a convergent subsequence to a continuous function $f_0$ in $[a.b]$.
We must use the Arzela-Ascoli Theorem to prove this. It looks trivial that the operator is bounded, but i'm having some troubles to show that the operator is equicontinuous, i don't know if i can just use something like $$|Tf-Tg|=\left|\int_a^{x}f(t)dt-\int_a^{x}g(t)dt\right| \leq \|f-g\|_0 \cdot (b-a)$$ and finish the question. Any help is welcome!