Almost-sure vs. pointwise convergence — a single exceptional point

Suppose the probability space $(\Omega,\mathcal A,P)$ is defined as follows: $\Omega=[0,1]$, $\mathcal A$ is a sigma field of subsets of $\Omega$ and P is a Lebesgue measure. Suppose a sequence $\{X_n,n\ge1\}$ of random variables is defined as $X_n(w)=e^w+w^n, n\ge1$ and a random variable X is defined as $X(w)=e^w$ on $(\Omega,\mathcal A,P)$. Which of the following is not true as $n\to\infty$?