This is a practice problem that is quite tricky, it is not graded and I am stuck at part ii), appreciate any help in advance
Let $f_n:[0,1]\rightarrow \mathbb{R}$ be a sequence of measurable functions and suppose $|f_n|\leq g$ for all $n$ and $g\in L^1$.$$F_n(t)=\int_0^t f_n(x)dx$$
i) each $F_n$ is continuous on $[0,1]$
ii) There is a subsequence $(F_{n_k})$ converging uniformly on $[0,1]$.
The first part is not that hard, I just argued the integral of the positve and negative parts of $f_n$ are both continous, therefore $F_n$ is continuous.
However, when it comes to part ii), I am not sure how to proceed, I am aware that every bounded sequence has a convergent subsequence, but constructing a uniform convergent subsequence has another degree of freedom and there seems no obvious way to do that, not even knowing the result of part i). Thank you again for your effort