16 - Cauchy Convergence Criterion

Cauchy Convergence Criterion

Let $(a_n)$ be a sequence then

$$(a_n)\text{ converges}⟺(a_n)\text{ is Cauchy}$$

Proof

Proof $⟹$ - Convergent Sequences are Cauchy

Proof $⟸$
Assume that $(a_n)$ is Cauchy
Then $(a_n)$ is bounded by Cauchy Sequences are bounded

By Bolzano-Weierstrass Theorem,

$$(a_n)\text{ has a convergent subsequence }(a_{n_r})$$

Then by $(a_n)$ converges