2. For every subset $A$ of $N=\{1,...,n\}$ , let $\mu(A)=\sum_{a \in A}a$. Then $0 \leq\mu(A) \leq n\mid A \mid$ because $A$ contains $\mid A \mid$ elements all less than or equal to $n$.
Now since $S$ has $2^k$ subsets, and there are at most $\mu(S) \leq kn$ values of $\mu (S')$ where $S'$ is any subset of $S$, and since we suppose $2^k > kn$, then two of the values $\mu (S_1), \mu (S_2)$ must coincide.