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 $$f$$)?