Could someone prove that $\mathbb{P}[\omega:\lim_{n\to\infty}X_n(\omega) = X(\omega)] = 1$ iff $\lim_{n\to\infty}\mathbb{P}[\omega:\sup_{k>n}|X_k(\omega) - X(\omega)|>\epsilon] = 0$ ? Here $\{X_n\}_{n=1,2,\cdots}$ is a sequence of random variables. Those two are equivalent definitions of a.s. convergence of random variables.
Two equivalent definitions of a.s. convergence of random variables.
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real-analysis
probability
convergence
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0[Related](http://math.stackexchange.com/q/1963380/). – 2016-10-12