Convergent Subsequence of Cauchy Sequence

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Theorem

Metric Space

Let $\struct {A, d}$ be a metric space.

Let $\sequence{x_n}_{n \in \N}$ be a Cauchy sequence in $A$.

Let $x \in A$.


Then $\sequence{x_n}$ converges to $x$ if and only if it has a subsequence that converges to $x$.


Normed Division Ring

Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring.

Let $\sequence{x_n}_{n \mathop \in \N}$ be a Cauchy sequence in $\struct {R, \norm {\,\cdot\,} }$.

Let $x \in R$.


Then $\sequence {x_n}$ converges to $x$ if and only if $\sequence {x_n}$ has a subsequence that converges to $x$.


Normed Vector Space

Let $\struct {X, \norm {\,\cdot\,} }$ be a normed vector space.

Let $\sequence{x_n}_{n \mathop \in \N}$ be a Cauchy sequence in $\struct {X, \norm {\,\cdot\,} }$.

Let $x \in X$.


Then $\sequence {x_n}$ converges to $x$ if and only if $\sequence {x_n}$ has a subsequence that converges to $x$.