40A05-CharacterizationOfConvergenceOfSequencesInMetricSpaces.tex

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\pmtitle{characterization of convergence of sequences in metric spaces}
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Let $(M, d)$ be a metric space, and let $ \mathbb{N} = P_{1} \bigcup \cdots \bigcup P_{k}$ be a partition of the set of natural numbers such that $P_{i}$ is infinite for every $i$, that is, there is a bijection $f_{i} \colon \mathbb{N} \to P_{i}$. Then, given a sequence $(x_n)_{n \in \mathbb{N}}$, it converges to $x \in M$ if and only if the subsequence $$(x_{f_i(n)})_n$$ converges to $x$ for every $i = 1, \cdots, k$.


\textbf{Examples}

If you have a sequence $(x_n)_n$ and a natural number $k$, and you know that it converges to $x$ for every corresponding subsequence over the classes of remainders modulo $k$, then it converges to $x$.
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