Does L^2 Convergance Imply Convergance of L^2 norms?

[logarithmic]

The answer seems to obviously be yes. But it's not so obvious to show it.

I'm working with random variables. So the L^2 norm of X is E(X^2)^{1/2}, where E is the expected value. Thus, we want to show: if E((X_n-X)^2)\to0, then E(X_n^2)\to E(X^2).

From E((X_n^2-X)^2)\to0, we get
E(X_n^2)\to2E(X_nX)-E(X^2).

I think it should be true that 2E(X_nX)\to2E(X^2), which would prove the result, but I'm not sure how to prove that.

Any help?

Or a reference? Is the result in a book?

 

[micromass]

Use the Cauchy-Bunyakovski-Schwarz inequality

|E(X_n X) - E(X^2)| = |E( (X_n - X) X)|\leq \sqrt{ E((X_n - X)^2) E(X^2)}