So I have a definition;(adsbygoogle = window.adsbygoogle || []).push({});

X_{n}n=1,2.... is a sequence of random variables on ( Ω,F,P) a probability space, and let X be another random variable.

We say X_{n}converges to X almost surely (P-a-s) iff P({lim_{n →∞}X_{n}=X}^{C}) = 0

It then goes on to say that checking this is the same as checking

lim_{m →∞}P({Sup_{n≥m}|X_{n}-X| ≥ε }) = 0

Can somebody please explain why this is true, I don't understand at all how to get from one to the other properly.

Thanks!

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# Convergence P-a-s

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