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

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

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

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

CB