Prove or disapprove

Question

Prove or disapprove:

Let $f:[1, +\infty)\rightarrow\mathbb{R}$ be a real function such that $f$ is increasing and continuous. Also, suppose

$\int_{1}^{+\infty}f(t)dt$ is convergent. Then, $f\geq 0$ or $f\leq 0$ on $[M, +\infty)$ for some $M\geq 1$.

Answer 1

Let $S:=\sup\{f(x): x\geq 1\}$. If $0<S<\infty$, then $f(x_{0})>0$ for some $x_{0}\geq 1$ and hence $f(x)>0$ for all $x\geq x_{0}$.

If $S\leq 0$, then $f(x)\leq S\leq 0$ and hence $f(x)\leq 0$ for all $x\geq 1$.

Assume that $S=\infty$, then $\lim_{x\rightarrow\infty}f(x)=\infty$, we can find some $M>0$ such that $f(x)>1$ for all $x\geq M$, then taking integrals both sides we have $\displaystyle\int_{M}^{\infty}f(t)dt\geq\int_{M}^{\infty}1dt=\infty$, a contradiction.

So we must have $S<\infty$.