On the sums of elements of uncountable sets

I want to prove the following proposition:

Given any uncountable set of real numbers S, there exists a countable sub-collection of numbers in S, whose sum is infinite.

Please point me in the right direction.

Related Calculus and Beyond Homework Help News on Phys.org
Office_Shredder
Staff Emeritus
Gold Member
As a small hint, consider for each n how many elements can have absolute value smaller than 1/n