Prove the existence of an uniformly convergent subsequence $ \{ f_{k_{j}} \}_{j \in \mathbb{N}} $ of the sequence $ \{ f_{k} \}_{k \in \mathbb{N}} $ in the following cases :
a) Each $f_{k} : [0,1] \rightarrow \mathbb{R}$ is a polynomial with degree $ < $ 101 and coefficients $a_{0,k} , a_{1,k} , \ldots a_{100,k}$ with $[a_{j,k}] \leq 1$
b) Each $f_{k} : [0,1] \rightarrow \mathbb{R}$ implies $f_{k}(0) = 0$, and is the antiderivative of a continuous function $g_{k}$ with $||g_{k}||_{C^{0}} \leq 1$
Do I have to think of the Arzelà–Ascoli theorem to solve this ? What is the best approach ?