Suppose $\{X_n\}$ is a sequence of $iid$ random variables with mean $0$ and variance $\sigma^2$. Then by weak law of large numbers, $\bar{X_n} \overset{p}{\to} 0$.
Consider the following application of continuous mapping theorem: define a function $g(x) = x^2$. Then $g(\bar{X_n}) \overset{p}\to g(0) = 0^2 = 0$. However, by weak law of large numbers, $\bar{X_n}\bar{X_n} \overset{p}\to E[X_nX_n] = \sigma^2$, since $E[X_iX_j] = 0$ for $i \neq j$ by independence.
What did I do wrong?