# Real analysis: Compact Sets

• March 7th 2010, 07:18 PM
Phyxius117
Real analysis: Compact Sets
Here is the problem:
http://i10.photobucket.com/albums/a1...s11787/335.jpg

I'm so hopelessly lost... Help Please!!!!
• March 7th 2010, 07:27 PM
Drexel28
Quote:

Originally Posted by Phyxius117
Here is the problem:
http://i10.photobucket.com/albums/a1...s11787/335.jpg

I'm so hopelessly lost... Help Please!!!!

A metric space is compact if and only if it's totally bounded and complete. I bet you know that $\mathbb{Q}$ is not totally bounded (under the usual metric), $\mathbb{Q}_{[0,1]}$ is not complete, $\mathbb{R}$ is not totally bounded

try d)

Also, consider that in a metric space (or in a more general scenario) if $x_n\to x$ then $C=\{x\}\cup\left\{x_m:m\in\mathbb{N}\right\}$ is compact. See if that helps