Is $[0,\infty)$ a closed subset?

Solution 1:

This set is indeed closed. Note that $+\infty$ is not a real number, sequences which tend to it are therefore non-convergent and have no limit in $\mathbb R$.

From this we can easily infer that $[0,\infty)$ is closed, since every sequence of positive numbers converging to a limit would have a non-negative limit which is in $[0,\infty)$.

Solution 2:

Note that the complement of $[0, \infty)$ is $(-\infty, 0)$, which is open in the usual topology on $\mathbb{R}$. Therefore $[0, \infty)$ is closed. I often find looking at the complement easier than thinking of limit points.