Let $ f:[a,\infty)\rightarrow \mathbb{R}$ be a uniformly continuous function in that range. $ \int_{a}^{\infty} f$ converges. Prove that $ \lim_{x\to\infty} f(x)=0$

Hint: Use the sequence $ F_n(x)=n\int_{x}^{x+\frac{1}{n}} f$ .

Honestly I have been trying to solve this one for some time but the hint really confuses me.

I have tried to mess around with $ F_n(x)$ a bit, for example by using the fundamental theorem but it still seems like such a random choice and I can’t make anything out of it.

Any guidance/explanations will be appreciated.

Please use the hint in the question.