----- > [!proposition] Proposition. ([[Cauchy sequence is convergent iff has convergent subsequence]]) > Let $(X,d )$ be a [[metric space]] and consider a [[Cauchy sequence]] $(x_{n})$ in $X$. Then $(x_{n})$ [[sequence|converges]] in $X$ if and only if it has a [[subsequence]] which [[sequence|converges]] in $X$. ^proposition > [!proof]+ Proof. ([[Cauchy sequence is convergent iff has convergent subsequence]]) > One direction is immediate. For the other, suppose the Cauchy sequence $(x_{n})$ has a subsequence $(x_{n_{k}})$ which converges in $X$, say, to $L \in X$. By construction, $(x_{n})$ converges in (say, to $x$) the [[completion of a metric space|completion]] $\overline{X}$ of $X$. Then $\big(x_{n_{k}}\big)$ converges to the same limit $x$ in $\overline{X}$. Since limits are unique, $L=x$, hence $x \in X$. > > (This is very high tech, does not need this fanciness) ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```