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Consider a sequence of random variable $\{X_n\}$. Let $$A_n = \sup\{|X_k - X_l|: k,l \geq n\}$$ $$B_n = \sup\{|X_k - X_n|: k \geq n\}$$

Now to prove a.s. convergence of $\{X_n\}$, I have seen in a theorem proof that $B_n \to^p 0$ is sufficient as that makes $A_n \to 0$ a.s.

My question is that is this technique useful to prove a.s. convergence. Are there many theorems where this technique is used ?

Thanks

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Maybe there are other examples, but the following one is quite classical: the almost sure convergence of the partial sums of an independent sequence with converges in probability. A proof is given in this thread.

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