Does $\lim\limits_{n\to\infty}|X_n-X|=0$ imply $\lim\limits_{n\to\infty}|X_n-X|^p=0\hspace{0.2cm}\forall p>1$?

Question

Let $\left(X_n\right)_{n\geq1}$ be a sequence of random variables and $X$ a random variable as well. Does $\lim\limits_{n\to\infty}|X_n-X|=0$ imply that $\lim\limits_{n\to\infty}|X_n-X|^p=0\hspace{0.2cm}\forall p>1$ as well?
That is, is it true that $$\lim\limits_{n\to\infty}|X_n-X|=0\Rightarrow\lim\limits_{n\to\infty}|X_n-X|^p=0\hspace{0.2cm}\forall p>1\;?$$

Answer 1

Yes, it is true. Given $p > 1$, the function $x \mapsto |x|^p$ is continuous.

Thus if we have a sequence $(a_n)_n$ in $\mathbb{R}$ with $\lim_n a_n = a$, then by continuity of $x \mapsto |x|^p$ also $\lim_n |a_n|^p =|a|^p$.

Hence, we can conclude.