Convergent Subsequence of Cauchy Sequence
Theorem
Metric Space
Let $\struct {A, d}$ be a metric space.
Let $\sequence {x_n}_{n \mathop \in \N}$ 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 $x$.