limiting distribution

Let X1,....,Xn be a random sample from a Poisson($\displaystyle \lambda$) distribution.

Find the limiting distribution of $\displaystyle \overline{X^2}$.

I found the mean to be E( $\displaystyle \overline{X^2}$) = $\displaystyle (1/n)*E(X_i)^2$

but then i don't know which method to use.

Thank you

The square is a continuous function.

Since $\displaystyle \bar X\to \lambda$

we have $\displaystyle \left(\bar X\right)^2\to \lambda^2$


and I don't understand what this.... $\displaystyle (1/n)*E(X_i)^2$ means
I guess zero as n goes to infinity