Sets of integers, different sums

There is a natural number http://www.mathhelpforum.com/math-he...21e8759df3.png. Prove that from any set consisting of integer numbers that has more than http://www.mathhelpforum.com/math-he...55a5f872b9.png elements we can take a subset S that has http://www.mathhelpforum.com/math-he...538d2ed552.png elements and:

For any two different subsets http://www.mathhelpforum.com/math-he...301a0f0ed1.png the sum of all elements of set http://www.mathhelpforum.com/math-he...b72eacbe29.png is different from the sum of all elements of set http://www.mathhelpforum.com/math-he...957afab571.png. (We assume that the sum of all elements of an empty set equals 0).

I would be really grateful if anyone could help me with this assignment.

I've already tried set theory and induction. Now I think this problem must have something to do with number theory, but this branch is so vast that I don't know where to start looking.