Let $S$ and $W$ be subsets of $\mathbb{R}$, with the usual metric,
\begin{align*} S &= \left\{\frac{1}{n} :n\in \mathbb{N}\right\}\cup\{0\} \\ W &= \left\{n+\frac{1}{n}: n\in\mathbb{N}\right\} \end{align*}
I have to check for completeness of these two subsets.
Here is my attempt:
$S$ being closed subset of $\mathbb{R}$ is complete.
$W$ is not closed in $\mathbb{R}$ since it doesn't contain all of its points and hence not complete.
I am not sure whether I am correct or not? Please help me with this.
Thanks