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I have a problem with following task: We have $X_1,X_2,...$ independent random variables, that almost surely $X_1\ge X_2\ge...\ge 0$. Prove that if $X_n\rightarrow0$ converges in probability then $X_n\rightarrow0$ converges almost surely. I am hitting a wall with this one. I will be very glad for as simple as possible explanation.

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  • $\begingroup$ As $X_n$ is decreasing a.s., it converges a.s. to the limit inferior. Convergence a.s. implies convergence in probability and limits are unique a.s. $\endgroup$ – user251257 May 23 '17 at 21:39
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Let $u_i$ be the essential supremum of $X_i$, i.e. the least constant $c$ such that $X_i \le c$ a.s. If $X_{i-1}$ and $X_i$ are independent but $X_{i-1} \ge X_i$ a.s., then $c_i$ must exist and $X_{i-1} \ge c_i \ge X_i$ a.s. Show that if $X_i \to 0$ in probability, $c_i \to 0$.

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