Which of the following metric spaces $X$ with the metric induced by usual Euclidean metric on $\Bbb R^n$ are complete?

Question

A. $X=\Bbb Z \times \Bbb Z \subset \Bbb R \times \Bbb R$

B. $X=\Bbb Q \times \Bbb R \subset \Bbb R \times \Bbb R$

C. $X=(-\pi,\pi) \cap \Bbb Q \subset \Bbb R$

D. $X=[-\pi,\pi] \cap (\Bbb R - \Bbb Q) \subset \Bbb R$

I choose option A as an answer because $\Bbb Z \times \Bbb Z$ is closed in $\Bbb R \times \Bbb R$ and $\Bbb R \times \Bbb R$ is complete.

$\Bbb Q$ and $\Bbb R- \Bbb Q$ are not closed in $\Bbb R$ hence options B,C,D are false.

Am I right?

Answer 1

A is true because there are no nontrivial Cauchy sequences! ($\mathbb Z\times\mathbb Z$ is discrete).

For $B$, $C$, and $D$, you should be able to easily demonstrate Cauchy sequences that do not converge.