With your statement, I'd say: the subsequences and have the same sum (and so do the empty subsequences). Do you want a strict subsequence? Writing your own steps could have helped to guess what you're looking for.

My guess (since it makes my proof work): your hypothesis should be and the numbers are non-zero (to avoid for all ). In this case, here is a solution: consider the differences , where and . There are of them. What about the possible values? is an integer, it is less than and greater than , so there are at most values for the 's. By the pigeonhole principle, there are two couples such that . For instance, (they can't be equal since it would imply that because the numbers are non-zero). Then and these subsequences aren't empty. qed