If f(x) increasing, f’(x) > 0.

dy=f’(x)dx, dx/dy=1/f’(x) > 0.

Therefore inverse function is increasing.

EDIT:

Given, if a>b, f(a)>f(b)

assume f(a)<f(b) but a>b. a>b → f(a)>f(b). Therefore b>a.

therefore, if f(a) > f(b), a>b

ie, inverse function is also increasing

shortened version from wicki-

Inverse of increasing function is increasing - Calculus

(Seems obvious now but never occurred to me.)