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.)