This question involves formal proof and set theory, as well as a knowledge of functions.

Suppose f is increasing. Then for all h>0, f(x+h)>f(x). Then by rules of inequalities, 1/f(x+h) < 1/f(x). So g(x+h)<g(x) where g=1/f. Hence g is decreasing.

Proof can be reversed to get iff.