Prove: Cauchy sequences are converging sequences

[Mathematicize]

Homework Statement

I want to prove that if a sequence a[n] is cauchy then a[n] is a converging sequence

Homework Equations

What I know is:

1. a[n] is bounded
2. any subsequence is bounded
3. there exists a monotone subsequence
4. all monotone bounded sequences converge
5. there exists a convergent subsequence by the above

 

[jbunniii]

Homework Statement

I want to prove that if a sequence a[n] is cauchy then a[n] is a converging sequence

Homework Equations

What I know is:

1. a[n] is bounded
2. any subsequence is bounded
3. there exists a monotone subsequence
4. all monotone bounded sequences converge
5. there exists a convergent subsequence by the above

I assume your sequence is real-valued, since you are talking about monotone subsequences.

What you have done so far is fine, assuming you have proofs for each claim. So now you need to prove that any Cauchy sequence which has a convergent subsequence must converge.

Let $L$ be the limit of the subsequence $(a_{n_k})$. Given $\epsilon > 0$, there exists a natural number $N$ such that $|a_{n_k} - L| < \epsilon$ for all $n_k \geq N$. There also exists a natural number $M$ such that $|a_n - a_m| < \epsilon$ whenever $n,m \geq M$. How can you combine these two pieces of information to conclude that $a_n$ converges to $L$?