If $f \in L^p(\mathbb{R})$ then $\lim_{x \rightarrow \infty} \int_x^{\infty} |f(t)|^pdt=0$

Question

If $f \in L^p(\mathbb{R})$ then $\lim_{x \rightarrow \infty} \int_x^{\infty} |f(t)|^pdt=0$ where $ p \in (1,\infty)$

I tried to use this Problem but if $x\to\infty$, also $a\to\infty$, so $\int_a^{\infty}|f(t)|^p dt$ is not a constant. So, I can no longer find how to proceed by any other means, could you help me with this?

Answer 1

I suppose you wrote $\int_x^{\infty} |f(x)|^{p}dx$ for $\int_x^{\infty} |f(t)|^{p}dt$.

This is an immediate consequenca of DCT. $I_{(a,\infty)} |f(x)|^{p}$ tends to $0$ at every point and it is dominated by the integrable function $|f|^{p}$. Hence its integral tends to $0$.