1. Let f and g be increasing functions such that the sets {f(1), f(2), . . . } and {g(1), g(2), . . . } partition the positive integers. Suppose that f and g are related by the condition  Prove that...

1. Let f and g be increasing functions such that the sets {f(1), f(2), . . . } and {g(1), g(2), . . . } partition the positive integers. Suppose that f and g are related by the condition
 Prove that

2. Do there exist real numbers α, β, and γ such that Spec(α), Spec(β), and Spec(γ) together partition the set of positive integers?