# characterization of convergence of sequences in metric spaces

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 $i=1,\cdots,k$.

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 $x$.

Title characterization of convergence of sequences in metric spaces CharacterizationOfConvergenceOfSequencesInMetricSpaces 2013-03-22 15:06:10 2013-03-22 15:06:10 gumau (3545) gumau (3545) 4 gumau (3545) Theorem msc 40A05 msc 54E35