Consider $\mathbb{R}$ with it's usual Euclidean metric. Show that the subspace $[a,b]$ is complete, with $a<b$ in $\mathbb{R}$
I know that for a subspace to be complete, every Cauchy sequence in it must converge in that subspace. How do I apply this to the question? Do I just point out that because $[a,b]$ is closed that it contains all of the limits of the sequences contained in it, and therefore complete?
Yes, that's basically it.
For classwork it would probably be good to add just a few more intermediate steps along the lines of
Suppose we have a sequence that's Cauchy in $[a,b]$. Since the metric on $[a,b]$ is a restriction of the metric on $\mathbb R$, the sequence is also Cauchy in $\mathbb R$, and therefore (since $\mathbb R$ is complete) it has a limit in $\mathbb R$. This limit must actually be in $[a,b]$ because ...
