In usual Euclidean metric on $\mathbb{R}^n$. Which of the following metric spaces X is complete

Question

For $X ⊂ \mathbb{R}^n$, consider $X$ as a metric space with metric induced by the usual Euclidean metric on $\mathbb{R}^n$. Which of the following metric spaces X is complete?

$A. X = \mathbb{Z} \times \mathbb{ Z} \subset \mathbb{ R} \times \mathbb{R}$

$B. X = \mathbb{Q } \times \mathbb{ R} \subset \mathbb{R} × \mathbb{R}$

$C.$ $X = (−π, π) \cap \mathbb{Q} \subset \mathbb{R}$

$D.$ $X = [−π, π] \cap (\mathbb{R} \setminus \mathbb{Q}) ⊂ \mathbb{R}$

I thinks option A,B and C will be correct As both $\mathbb{Q}$ and $\mathbb{R}$ are complete

Is its correct ?

Any hints/solution will be appreciated

thanks u

Answer 1

Actually, $\mathbb Q$ is not complete and answer B. is wrong. Only answer A. is correct. I hope that you know how to justify that.