# Convergence of Random Variables in mean

If $$E[|X_n-X|^r]\rightarrow0$$ prove that $$E|X_n^r|\rightarrow E|X^r|$$
for every $r\ge 1$

This is the very notation used. I believe it should be:
$$E[|X_n|^r]\rightarrow E[|X|]^r$$

Attempt I think I can obtain $E[X_n]\rightarrow E[X]$ using Jensen Inequality but I don’t think this helps. I have no further idea.

#### Solutions Collecting From Web of "Convergence of Random Variables in mean"

For any norm $\|\cdot\|$, by the triangle inequality,
$$\|X_n\|\le\|X_n-X\|+\|X\|$$
and
$$\|X\|\le\|X-X_n\|+\|X_n\|$$
so that
$$|\|X_n\|-\|X\||\le\|X_n-X\|.$$
This property is sometimes called the continuity of the norm.

$(\operatorname E|X|^r)^{1/r}$ is the norm of the space of random variables with $\operatorname E|X|^r<\infty$ for $r\ge1$. We obtain that
$$|(\operatorname E|X_n|^r)^{1/r}-(\operatorname E|X|^r)^{1/r}|\le(\operatorname E|X_n-X|^r)^{1/r}.$$