# Convergence in distribution implies convergence in probability

Let $$X_n$$ be a sequence of random variables in the probability spaces $$(\Omega_n, \mathcal{A}_n, P_n )$$ such that $$X_n$$ converges to $$N(0,\sigma^2)$$ in distribution. Does it imply that the sequence $$\exp(X_n)$$ converges to $$1$$ in probability? If yes, how can I show it?

If $$\exp(X_n)\to 1$$ in probability then since $$\log$$ is a continuous function $$X_n\to 0$$ in probability. Contradicition.