Show that $[1, \infty) \subset \mathbb{R}$ is complete

Question

Assume $\mathbb{R}$ is equipped with standard Euclidean metric, I need to show that $[1, \infty)$ is a complete space, when $[1,\infty)$ is viewed as a subspace of $\mathbb{R}$. However, the complete subsets of a complete space are exactly those subsets that are closed, how can $[1,\infty)$ be complete if $[1,\infty)$ is not closed?

Answer 1

The complement is open, therefore the set is closed.

Answer 2

Note: There is no contradiction since $[1,\infty)$ is closed in $\mathbb{R}$.

Your proof would then look something like this:
Let $[1,\infty)\subset \mathbb{R}$. We know that $\mathbb{R}$ is a complete metric space, and also that $[1,\infty)$ is closed, because its complement $(-\infty,1)$ is open. Since a closed subset of a complete metric space is complete (see here for a proof), it follows that $[1,\infty)$ is a complete subspace of $\mathbb{R}$.