Define a function w:Z×Z→Z as follows. For ∣a∣,∣b∣≤2, let w(a,b) be as in the table shown; otherwise, let w(a,b)=0.aw(a,b)−2−1012−2−1−22−2−1−1−24−44−2b02−412−421−24−44−22−1−22−2−1
For every finite subset S of Z×Z, define A(S)=(s,s′)∈S×S∑w(s−s′). Prove that if S is any finite nonempty subset of Z×Z, then A(S)>0. (For example, if S={(0,1),(0,2),(2,0),(3,1)}, then the terms in A(S) are 12,12,12,12,4,4,0,0,0,0,−1,−1,−2,−2,−4,−4.)