A sequence is convergent if and only if the sequence is a Cauchy sequence.ᐥ
If ｢a⸤n⸥｣ is a convergent sequence, then for all ε>0 there exists a natural number N such that n,m>N ⇒
|a⸤m⸥−a⸤n⸥|
= |a⸤m⸥−L+L−a⸤m⸥|
≤ |a⸤m⸥−L|+|L−a⸤m⸥|
┄
If ｢a⸤n⸥｣ is convergent, for all ε>0 there exists a natural number N such that n,m>N ⇒ |a⸤m⸥−L|<ε/2 and |a⸤n⸥−L|<ε/2.
┄
< ε/2+ε/2
= ε.
This means a Cauchy sequence.
So a convergent sequence is a Cauchy sequence.
If ｢a⸤n⸥｣ is a Cauchy sequence, then there exists a natural number N such that n>N ⇒ |a⸤n⸥−a⸤N⸥|<1.
This means that ｢a⸤n⸥:n>N｣ is bounded.
Since ｢a⸤n⸥:n≤N｣ is finite, ｢a⸤n⸥｣ is bouded too.
By Bolzano-Weierstrass theorem, ｢a⸤n⸥｣ contains a convergent subsequence.
Let the subsequence and its limit be ｢s⸤i⸥｣ and L respectively.
Then for all ε>0 there exists a natural number N such that i,n>N ⇒
|a⸤n⸥−L|
= |a⸤n⸥−s⸤i⸥+s⸤i⸥−L|
≤ |a⸤n⸥−s⸤i⸥|+|s⸤i⸥−L|
┄
For all ε>0 there exists a natural number N such that i>N ⇒ |s⸤i⸥−L|<ε/2.
For all ε>0 there exists a natural number N such that n,i>N ⇒ |a⸤n⸥−s⸤i⸥|<ε/2.
┄
< ε/2+ε/2
= ε.
So a Cauchy sequence is convergent.
