Prove that $\{(0,0)\}\cup \{(x,\sin(1/x))|x>0\}$ is complete in $\mathbb R^2$.

Question

I am trying to show that $\{(0,0)\}\cup \{(x,\sin(1/x))|x>0\}$ is complete in $\mathbb R^2$ with the Euclidean metric.

I know I just need to show it is closed, which seems rather obvious but I am not sure how I would set out a rigorous proof of this fact.

Answer 1

$\left(\frac 2 {(4n+1)\pi}, \sin \left(\frac {(4n+1)\pi} 2\right)\right) \to (0,1)$ so this is a cauchy sequence in the set which is not convergent.