How to modify the conditions so that the theorem is true. 
Let $f(x)$ and $g(x)$ be two increasing and differentiable functions from $\Bbb{R}$ to $\Bbb{R}$. If $f'(x)>g'(x)$ $\forall x$ then there exists an interval $[a,\infty)$ for some real number $a$ for which $f(x)>g(x)$.

This theorem is false, as shown by many counter examples. But how do I modify the conditions so that the theorem becomes true.?

No, this isn't true. $g$ could be a line, for instance, and $f$ could increase to it as an oblique asymptote. You need $f'>g'+\epsilon$ for some positive $\epsilon$, in which case the fundamental theorem of calculus will get you the result right away.
