TITLE: $\Gamma \subset \mathbb{R}^{+}$ is uncountable. Can we choose a sequence from $\Gamma$ of which the sum is $\infty$
QUESTION [8 upvotes]: If $\Gamma$ is a set of uncountably many different positive real numbers, can we choose a sequence of pairwise different positive numbers from $\Gamma$, say $\{a_n\}$, such that $\sum a_n = \infty$ ?

I am considering such problem and I think the answer should be yes. But what's the rigorous way to prove it? Please give me some hints and I will appreciate any help :-)

REPLY [10 votes]: If not, then there are only finitely many elements in $[\epsilon,\infty)$, for any $\epsilon>0$, and $\Gamma = \bigcup_{n=1}^\infty \left(\Gamma\cap[1/n,\infty)\right)$ is the countable union of finite sets.