Suppose that function f : R → R satisfies the inequality:
n
| ∑ 3^r{f(x + ry) - f(x - ry)}| ≤ 1 for every positive integer r, x, y
r = 1
Prove that f(x) = constant
If for all , then f is constant.
Suppose f is not constant. Then there exists an a,b such that , or for some . Choose a positive integer n such that . Define x,y such that and
Now, by construction, , for the last term of the summation. Now we have two cases:
Case 1: so f does not meet the hypothesis for the choice n,x,y as prescribed.
Case 2: or, the terms for counterbalanced to put the whole sum back under 1.
More rigorously, . WLOG, suppose . Then . So f does not meet the hypothesis for the choice n-1,x,y as prescribed.
Ergo, by contrapositive, the theorem is true.