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Convergence in probability

From $X_n\to X$ in probability, choose $n_k$ with
$\mathbb P(|X_{n_k}-X|>2^{-k})<2^{-k}$. The first Borel--Cantelli lemma then gives $X_{n_k}\to X$ almost surely.

Almost-sure subsequence from convergence in probability

Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 26H / a / Solution

Almost-sure subsequence from convergence in probability

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Almost-sure subsequence from convergence in probability

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