[Math] Limit of Cauchy Sequence


Let $(X,d)$ be a metric space and $(x_n)$ be a Cauchy sequence in $X$.
Is there a limit $x$ for $(x_n)$ whether $x$ in X or not ?

In general, does every Cauchy sequence has a limit, if that limit point lies in the same metric space which Cauchy sequence defined, then that Cauchy sequence converges otherwise it's divergent?

Best Answer

A Cauchy sequence in a metric space $(X,d)$ is not necessarly convergent. for example, consider the metric space $(\mathbf{Q},d)$ where d is the usual absolute value. And consider the sequence $x_{n}$, $x_{1}=1$, $x_{n+1}=\frac{1}{2}(x_{n}+\frac{2}{x_{n}})$. One can check that $x_{n}$ is a Cauchy sequence in $\mathbf{Q}$ but has no limit in $\mathbf{Q}$.

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