Let X be a random variable. Let Xn be a sequence of random variables.

Then E(|Xn - X|) -> 0 as n->∞ implies E(Xn) -> E(X) as n->∞.

[I can prove this by |E(W)|≤E(|W|) and squeeze theorem.]

Is it also true thatE(Xn) -> E(X) implies E(|Xn - X|) -> 0??

How can we prove it?

Any help is much appreciated!