Proving that if $\int_D|\varphi_n|^q\to0$ and $1<p<q<\infty$, then $\int_D|\varphi_n|^p\to0$

Question

I have the following situation : let $1< p < q < \infty$ . Consider in $\mathbb R^n$ the Lebesgue measure. Let $\{\varphi_n\}$ a sequence of functions in $C^{\infty} (D)$ ( $D$ a open subset of $\mathbb R^n$ not necessarialy bounded ) . Suppose $\int_{D} |\varphi_n |^q\rightarrow 0$.

I showed that (using Holder inequality) $\int_{G} |\varphi_n |^p\rightarrow 0$ where $G$ is a arbitrary subset of $D$ with the closure of G compact in R^n. Someone can give me a suggestion how to prove that

$$\int_{D} |\varphi_n |^p\rightarrow 0?$$

Thank you

Answer 1

If $p=2$, $q=4$, $D=(1,\infty)$ and $\varphi_n(x)=\dfrac{1}{n\sqrt{x}}$, you will have $\int_D|\varphi_n|^4\to 0$ while $\int_D|\varphi_n|^2=\infty$ for all $n$.