I thought about this for a while and my lecturer gave me the following result to prove the uniqueness of the point of the intersection.
Let F and G be function defined as
Suppose there exist x,y with such that
Then there exists a unique such that
Let 0,\infty) \rightarrow [0,1]" alt=" H0,\infty) \rightarrow [0,1]" /> be a function defined by
and let WLOG .
It follows that H is continuous since it is the difference of continuous functions F and G.
Then by (1) and (2), we have
Then it follows from the Bolzano's Intermediate Value Theorem that there exists a such that
which is equivalently,
I think this point is unique in the interval .
I'm not so sure about this because although I have proved that there is a point of intersection, yet the rate of increase for each function changes over different values for . I do not think the assumption (3) guarantees that G never over takes F again since we can have for some value z that is greater than y such that .