On the sums of elements of uncountable sets

[bbkrsen585]

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.

 

[Office_Shredder]

As a small hint, consider for each n how many elements can have absolute value smaller than 1/n