2
$\begingroup$

a) Show that $N$ = {1,2,...} is closed in $\mathbb R$

b) Show that $Z$ $\times$ $Z$, where $Z$ = {0, $\pm$1, $\pm$2,...} is closed in $\mathbb R^2$

I am not sure how to prove this homework question, should I start with claiming that $N$ is closed if $R$ \ $N$ is open. Could you help me to prove these two?

  • 1
    Take a convergent sequence $u_{n}$ in $\mathbb{N}$ or $\mathbb{Z}\times\mathbb{Z}$. Try to show that exist some $n_{0}\in \mathbb{N}$ such that $u_{n}$ is constant for n>n_{0}.2012-10-15

1 Answers 1

2

a) What is the complement of $\,\Bbb N\,$ in $\,\Bbb R\,$? Can you write it down as some union of open subsets?

b) Almost identical as (a) above but within $\,\Bbb R^2$