I’m trying to prove that $(mathbb{N},d)$ is a complete space where $d=left | m-n right |$.
So I define $a_{n}:=n$ , if it’s cauchy we know that $forall epsilon$>0 there exist $Nin mathbb{N}$ s.t for any $m,n>N$
$left |a_{n}-a_{m}right|<epsilon $
In order to show that $mathbb{N}$ is complete we have to show that $a_{n}$ is convergent in $mathbb{N}$.
Suppose that $lin mathbb{N}$
we want to show that
$left |a_{n}-lright|<epsilon$ $(*)$
We know that $a_{n}$ is cauchy so it’s bounded and it has a convergent subsequence $a_{n_{k}}$ which is convergent and lets say that $a_{n_{k}}rightarrow l$
From $(*)$ we have
$left |a_{n}-lright|<left|a_{n}-a_{n_{k}}right|+left|a_{n_{k}}-lright|<epsilon$
My approach is correct ?? because I’m confused and I’m not 100% sure.
Answer
To show a metric space is complete, you have to show that every Cauchy sequence converges to some limit point. So I’m going to take issue with “I define $a_n := n$.” Instead you should let $a_n$ be an arbitrary Cauchy sequence.
I’m also going to take issue with “[$a_n$] has a convergent subsequence $a_{n_k}$.” We don’t know that $a_n$ has such a convergent subsequence without doing a bit of extra legwork –in fact, any Cauchy sequence with a convergent subsequence must converge! (Exercise: Prove this.)
So how to fix the issues? You need to start with a Cauchy sequence $a_n$, which you know has for every $epsilon > 0$ there exists some $N$ so for $i,j > N$ we have $|a_i – a_j|<epsilon$. Given this sequence, you need to find some $linmathbb{N}$ such that $a_nto l$.
So we need to understand $mathbb{N}$ well enough that we can guess what $l$ should be. Say we have such a sequence of positive integers. What do you know about the distance between integers? If two integers are $epsilon$ away from each other, what can we say about them?