Issue with a simple positive function while using Reduce and FindInstance

Questions
1

The following function is positive, easy to see it, but Mathematica fails! Why? I have tried with Reduce, and Find Instance or the following. If one sets some parameters, such as T = 60 and S = 20 (in fact T > S) the <=0 of the function is False. But why Mathematica fails to justify analytically that it as >=0?:

Assuming[T > S && r > g && g > 0 && S > 0, 
 FullSimplify@
  Reduce[(E^((g - r) S + (g - r) T) (-S + E^((-g + r) T) S + T - 
       E^((-g + r) S) T))/((1 - E^((g - r) T)) (E^((g - r) S) - 
       E^((g - r) T))) > 0]]