Prove/Show that Limit is equal to 0.

I'm really struggling on this question. I've been thinking about it for awhile now but this is one of those where I don't really have much intuition on what to do.

Problem 3. Prove that if the limit $\lim_{x\to +\infty}f(x) =: L$ exists (finite or infinity) and the improper integral. $$\int_a^{+\infty}f(x) dx$$ is convergent, then $L = 0$.

Here is some facts that I know that I think I should perhaps use.

Facts/Knowledge I know: The above integral can be rewritten as:

$$\int_a^{\infty}f(x)dx = \lim_{A\to \infty}\int_a^{A}f(x)dx$$

We know that limit as x goest to infinity of $f(x)=L$. I need to somehow show that limit L is equal to 0. If I take the integral I get $\lim_{A \to \infty} F(A)-f(a)$ and I know this does not equal infinity as the integral converges. I'm not sure if this is on the right track or what to do next.

I was thinking maybe perhaps instead I could use the Cauchy Criterion to come up with a proof. Note: I'm not used to doing proofs with Improper integrals.

Theorem 1 (Cauchy Criterion). The improper integral (1) converges if and only if for every $\epsilon > 0$ there is an $M\geqslant a$ so that for all $A, B \geqslant M$ we have

$$\Bigg|\int_A^B f(x) dx\Bigg| < \epsilon$$

Assume that $L\in(0,\infty)$, then for some $M$, we have $f(x)>\dfrac{L}{2}$ for all $x\geq M$, then $\displaystyle\int_{M}^{\infty}f(x)dx\geq\int_{M}^{\infty}\dfrac{L}{2}dx=\infty$.

If $L\in(-\infty,0)$, then for some $N$, we have $f(x)<\dfrac{L}{2}$ for all $x\geq N$, then $\displaystyle\int_{N}^{\infty}f(x)dx\leq\int_{N}^{\infty}\dfrac{L}{2}dx=-\infty$.

Similar reasoning applies for the case that $L=\infty$ or $L=-\infty$. For example, $L=\infty$: Simply choose $M'$ such that $f(x)>1$ for all $x\geq M'$, the integrating both sides on $[M',\infty)$ to get the contradiction.