Cauchy Sequences

A Cauchy sequence is a sequence $\{x_n\}$ in a metric space $X$ with the property that as the sequence progresses, its terms get arbitrarily close to one another.

$$ \forall \varepsilon > 0, \exists N \in \N : \forall m,n \geq N , d(x_n, x_m) < \varepsilon $$

Note that this provides a notion of convergence, independent of limits. Moreover, convergence in the usual sense is heavily implied by being Cauchy, with the only caveat being that the ambient metric space may not contain its limit.

Convergent Sequences are Cauchy. If $\{x_n\}$ converges, it is Cauchy.

Proof </aside>

Cauchy in Compact $X$ Implies Convergence. If $X$ is compact and $\{x_n\}$ is Cauchy, then $\{x_n\}$ converges.

Proof </aside>

Cauchy in $\R^k$ Converges. If $X = \R^k$ $X = \R^k$$\{x_n\}$ is Cauchy, then $\{x_n\}$ converges.

Proof </aside>