When considered as a subset of [itex]\mathbb{R}^2[/itex], [itex]\mathbb{Z}[/itex] is a closed set.(adsbygoogle = window.adsbygoogle || []).push({});

Proof.

We will show, by definition, that [itex]\mathbb{Z} \subset \mathbb{R}^2[/itex] is closed.

That is, we need to show that, if [itex]n[/itex] is a limit point of [itex]\mathbb{Z}[/itex], then [itex]n \in \mathbb{Z}[/itex].

I think this becomes vacuously true, since our hypothesis is false, i.e. because [itex]\mathbb{Z}[/itex] has no limit points. Is this true, or am I just being silly?

Thank you!

Edit: (I know this can be proved, again by definition, by showing that [itex]\mathbb{R}^2 - \mathbb{Z}[/itex] is open.)

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# Set of integers is closed.

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