# countably many intersection of sets

• Mar 26th 2010, 09:33 PM
willy0625
countably many intersection of sets
How would you rigorously show
$\cap_{n=1}^{\infty}[a-\frac{1}{n},\infty)= (a,\infty)$ for each $a\in\mathbb{R}$
• Mar 26th 2010, 09:49 PM
Drexel28
Quote:

Originally Posted by willy0625
How would you rigorously show
$\cap_{n=1}^{\infty}[a-\frac{1}{n},\infty)= (a,\infty)$ for each $a\in\mathbb{R}$

Clearly $[a,\infty)\subseteq \left[a-\tfrac{1}{n},\infty\right),\text{ }\forall n\in\mathbb{N}$. Now, suppose that $b. We clearly have that $b0$ and thus by the Archimedean principle there exists some $m\in\mathbb{N}$ such that $\frac{1}{m}. It follows that $b\notin\bigcap_{n=1}^{\infty}\left[a-\tfrac{1}{n},\infty\right)$