Let $A$ be a bounded subset of $C_{[a,b]}$. I want to show that the set of functions of the form $\int_a^x f(t) dt$ for $f(t)\in A$ is compact (this problem appears in Kolmogorov & Fomin's Introduction to Real Analysis, page 107). I'm pretty sure I've successfully shown that the set is both uniformly bounded and equicontinuous. If the set is also closed, then it will follow that it is compact.
How can I go about showing that this set is closed? Am I going about the original problem the completely wrong way? If so, could someone point me in the right direction please? Thank you for any help.