[SOLVED] convergence in probability

Hi,

I've been banging my head against the wall for too many hours and would love a hint... I'm looking to show that

$\displaystyle

\frac {1} {n} \sum {(x_i - \overline{x_n})*(y_i - \overline{y_n})} \rightarrow Cov(X,Y)

$

(in probability), where (X_i, Y_i) are iid random vectors (finite 4th order moments).

I'm trying to use the definition of convergence in probability and get

$\displaystyle

P[|\frac {1} {n} \sum {(x_i - \overline{x_n})*(y_i - \overline{y_n})} - Cov(X,Y)| > \epsilon)]

$

$\displaystyle

\leq E[\frac {1} {n} \sum {(x_i - \overline{x_n})*(y_i - \overline{y_n})} - Cov(X,Y)]^2 / \epsilon^2

$

I'd appreciate a hint. I am not getting the Var(X,Y) part, I think, and I don't think it should be that hard!?

Thank you!