Are these metrics complete?

Question

Determine if these subsets of R are complete with the Euclidean metric?

a) $[0,\infty)$

b) $(0,\infty)$

I know the definitions of completeness and I know the Euclidean metric, but don't know how to test this...

Answer 1

Hint: $[0,\infty)$ is a closed subspace of $\mathbb{R}$, which is complete, and for $(0,\infty)$, take the sequence $\{1/n\}_{n\geq 1}$, which you can show to be Cauchy.

Theorem: Let $(M,d)$ be a complete metric space with metric $d$, and let $(S,d)$ be a subspace of $(M,d)$. Then if $S$ is closed in $M$, then $(S,d)$ is complete."

Proof:

Suppose $S$ is not complete, so there exists a Cauchy sequence $\{s_n\}_{n\in \mathbb{N}}$ in $S$ such that its limit, say $s$, in $M$ (which exists because $M$ is complete) does not lie in $S$. But then for every $\epsilon>0$ there exists an $N\in \mathbb{N}$ such that for all $n\geq N$ we have $d(x,x_n)<\epsilon$, so that there exists no $\epsilon$-ball containing $x$ and lying entirely in $M\setminus S$. Thus, $M\setminus S$ is not open, and consequently $S$ is not closed, giving us a contradiction. QED