Let f : [a,b] → R be a bounded function. Let c,d ∈ (a,b) such that c S = {a,c, b}, a partition of [a,b], and its refinement P = {a,c,d,b}. Prove:
L(f, S) ≤ L (f, P) ≤U(f, P) ≤ U (f, S)