\$mathbb{N}\$ complete metric space

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.