Prove if g is integrable, so is f

Suppose f and g are functions defined on the rectangle R and suppose there is a constant K such that
|f(x)-f(y)| <= K|g(x)-g(y)| for all x,y in R. Prove that is g is integrable on R, then so is f.

Since g is integrable on the rectangle R, given ε > 0, there exists a partition P of R such that
Σ(j) (g(x*,y*) - g(x^, y^)) ΔA_j < ε/K.

[Here, (x*, y*) denotes the point in a mesh from P which is the supremum of f, while
(x^, y^) denotes the point in a mesh from P which is the infimum of f. The index j runs through all meshes in P.]

Hence, Σ(j) (f(x*,y*) - f(x^, y^)) ΔA_j
≤ Σ(j) K(g(x*,y*) - g(x^, y^)) ΔA_j, by hypothesis
= K Σ(j) (g(x*,y*) - g(x^, y^)) ΔA_j
< K(ε/K)
= ε.

Hence, f is also integrable on R.

I hope this helps!