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


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.

If $f$ is bounded and $\lim_{x\to+\infty} F(x)=0$ then $\lim_{x\to+\infty} f(x)=0$


Let $ f:(0, + \infty) \rightarrow \mathbb{R}$ be a derivative function and $ F$ one of its primitives. Prove that if $ f$ is bounded and $ $ \lim_{x\to+\infty} F(x)=0$ $ then $ $ \lim_{x\to+\infty} f(x)=0$ $

What I have tried:

I observed that $ $ (f(x)F(x))’=f'(x)F(x)+f^2(x)$ $ It is obvious that the left term is $ 0$ when $ x\rightarrow+\infty$ . The problem is that we don’t know that $ f’$ is also bounded. If that would be the case, then $ f'(x)F(x)\rightarrow 0$ , so $ f\rightarrow 0$ .

Can you please help me solve this problem? Is it even correct or should $ f’$ be bounded(instead of $ f$ )?