Let: $f: [a , \infty)\to \mathbb{R}$ such that $\int_a^\infty f$ converge. Prove that $F(x) = \int_a^x f$ is a uniformly continuous

Question

Can someone help me proving this problem?
Let $f :[a , \infty)\to \mathbb{R}$ such that $\int_a^\infty f$ converge. Prove that $F(x) = \int_a^x f$ is a uniformly continuous function in $[a,\infty)$. Thanks!

Answer 1

Note that $F(x)$ is continuous in $[a,+\infty)$ and the $\lim_{x\to+\infty}F(x)$ exists and it is finite.

Then use Given a continuous function with an asymptote, prove that the function is uniformly continuous.