Two equivalent definitions of almost sure convergence of random variables.

Let $\{X_n\}_{n=1,2,\cdots}$ is a sequence of random variables. There are equivalent definitions of almost sure convergence of random variables. How can one prove the equivalence?

$\mathbb{P}[\omega:\lim_{n\to\infty}X_n(\omega) = X(\omega)] = 1 \Leftrightarrow \lim_{n\to\infty}\mathbb{P}[\omega:\sup_{k>n}|X_k(\omega) - X(\omega)|>\epsilon] = 0$