# \$mathbb{N}\$ complete metric space Code Answer

Hello Developer, Hope you guys are doing great. Today at Tutorial Guruji Official website, we are sharing the answer of \$mathbb{N}\$ complete metric space without wasting too much if your time.

The question is published on by Tutorial Guruji team.

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.

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?