I have been working on the following problem.
Prove that if a sequence $X_n$ of random variables satisfies $$\lim_{n,m\rightarrow \infty}P\left\{ \sup_{m<k\leq n} |X_k-X_m|\geq \delta \right\}=0 \hspace{1cm}\forall \delta>0$$ then there is a limiting random variable $X$ such that $X_n \xrightarrow[n\rightarrow \infty]{a.s}X$.
First of all, this question has been asked in "From Cauchy in Measure to Almost Sure Convergence" but the answer is not complete. Especially, it is not clear where he used the sup in the condition.
Opinion
At first glance, it seems right to start from noting that $X_n$ is Cauchy so there is a subsequence $X_{n_k}$ which is convergent a.s. to a random variable $X$. Then I was trying to use the argument given in the answer of the question linked above by showing that $$P\left\{\limsup_{n\rightarrow \infty}|X_n-X|>\lambda \right\}=0$$ for any $\lambda >0$. And note that
$$P\left\{\limsup_{n\rightarrow \infty}|X_n-X|>\lambda \right\}= P\left\{\lim_{n\rightarrow \infty}\sup_{n<k}|X_k-X|>\lambda \right\}$$ But seems there is no relation to the given condition and I don't know how to get further.
I thank in advance for any help!