Set Theory: Show A is finite or countably infinite

• September 12th 2011, 10:15 AM
jsmith90210
Set Theory: Show A is finite or countably infinite
I've been staring at this proof for several days now, and I have no idea where to even begin

Let A be a set of positive real numbers such that
Sigma: x <= 1
^any finite subset of A

Show A is finite or countably infinite

Any help is much appreciated
• September 14th 2011, 05:26 AM
CaptainBlack
Re: Set Theory: Show A is finite or countably infinite
Quote:

Originally Posted by jsmith90210
I've been staring at this proof for several days now, and I have no idea where to even begin

Let A be a set of positive real numbers such that
Sigma: x <= 1
^any finite subset of A

Show A is finite or countably infinite

Any help is much appreciated

Consider the sets: $S_n=\{a\in A: a>1/n\}\subseteq A$ where $n \in \mathbb{N}$

$S_n$ is clearly finite, and $A=\bigcup_{n\in \mathbb{N}}S_n$.

So what do we know about countable unions of finite sets?

CB