# N is closed?

1. Mar 23, 2006

We defined the definition of a closed set to be:

"$$F\subset\mathbb{R}$$ is closed if the limit of any convergent sequence in F is an element of F."

Now we have also defined that a sequence may "converge to infinity". Is infinity considered a point in N?

2. Mar 23, 2006

### Hurkyl

Staff Emeritus
$\pm\infty$ are elements of the extended real numbers, not the real numbers. Converging to infinity only makes sense when working within the extended reals.

So if you're working over the reals, then "converge to infinity" doesn't make sense, because the reals have no such element. A sequence that would converge to (plus) infinity in $\bar{\mathbb{R}}$ is not a convergent sequence in R.