Adem recently sent me an email claiming the following:

"Take a set of n+1 positive integers which sum to 2n. It is always possible to form a subset of those integers with sum n."

For instance, {1, 1, 1, 2, 3} is a set of 5 integers which sum to 8. I can form the subset {1, 3} which has a sum of 4.

Is Adem right? Can you prove his claim, or find a counterexample?