Generating functions in a proof

Question

I'm working at the moment with queues. In a proof of a theorem, there is the following statement:

Suppose $G(s)$ is a generating function and we have the following conditions: $$0<G(0)<1,$$ $$G(1)=1,$$ $$G'(1)>1.$$ Then there exists a number $b \in (0,1)$ sucht that $G(b)=b.$

There is written, that one should draw a picture and then it is clear... For me that is not so easy, how can I draw a picture with the conditions above? It's impossible for me to think of a drawing. Can anybody help me please?

I would really appreciate any idea or help!! Thank you!!

Answer 1

Since $G'(1) > 1$, the generating function must somewhere cross the line $y = x$

Sorry for the ugly picture!!!

edit: i am not familiar with queueing-theory, but i suppose the generating function is on the form $\sum a_nx^n$, i.e. continuous