How to prove or disprove for the statement?

Question

Let $f$ be a measurable function on $\mathbb{R}$ and $p\in[1,\infty)$. By Lebesgue Dominated Convergence Theorem, we know the statement " If $f\in$$L^p(\mathbb{R})$, then $\lim_{n\to\infty}\int_{n}^{n+1}|f(x)|^p\mathrm{d}x=0$ " is true. How about " If there is a $k>0$ with $f\in$$L^{p+k}(\mathbb{R})$, then $\lim_{n\to\infty}\int_{n}^{n+1}|f(x)|^p\mathrm{d}x=0$ " ? Is true or false ?

Answer 1

Notice that

$0\le\int_{n}^{n+1}\lvert f(x)\rvert^{p}dx\le\left(\int_{n}^{n+1}1dx\right)^{1-\frac{p}{p+k}}\left(\int_{n}^{n+1} \lvert f(x)\rvert^{p+k}\right)^{\frac{p}{p+k}}=\left(\int_{n}^{n+1}\lvert f(x)\rvert^{p+k}dx\right)^{\frac{p}{p+k}}$

so now we can apply squeeze theorem and the previous result.