For $f\in L^2(\mathbb R)$, is it true that $\lim_{n\to\infty} \int_{n}^{n+1}f(x)dx=0$?

Question

This is from a prelim in analysis that I took yesterday. We have $f\in L^2(\mathbb R)$, and we are asked to prove or disprove the following: $$ \lim_{n\to\infty} \int_n^{n+1} f(x)dx=0 $$ I believe that the statement is true. I attempted a proof which I've tried to recreate below. Any criticisms or improved solutions would be appreciated.

Answer 1

The statement is true.

We have: \begin{align*} &\int_\mathbb R |f(x)|^2dx <\infty\\ \implies& \sum_{n=1}^{\infty}\int_n^{n+1}|f(x)|^2dx<\infty\\ \implies& \lim_{n\to\infty}\int_n^{n+1}|f(x)|^2dx=0\\ \implies& \lim_{n\to\infty}\left|\int_n^{n+1}f(x)dx\right|^2=0&\text{(Cauchy-Schwarz)}\\ \implies& \lim_{n\to\infty}\int_n^{n+1}f(x)dx=0 \end{align*}