Bernoulli's Rule: Existence of a Point c in (a,b)

[Bachelier]

In his treatment of L'hôpital/Bernoulli's rule (please see attached), Rudin before ineq. $(17)$ mentions that since the differentiable quotient

$\frac{f'(x)}{g'(x)} \rightarrow A$ as $x \rightarrow a$ and $A<r$ then there exists a pt $c \in (a,b) \ s.t. \ a<x<c \Rightarrow \ \frac{f'(x)}{g'(x)}<r$

Is it so because $x$ approaches $a$ that's why he used $a<x<c$ instead of $c<x<b$

and why this $c$ in the first place? What's wrong with just saying, $\exists x \in (a,b)$ etc

 

[mathman]

The theorem is not there exists an x in the interval, but rather for all x, a < x <c, f'/g' < r.

 

[Bachelier]

The theorem is not there exists an x in the interval, but rather for all x, a < x <c, f'/g' < r.

True. Thanks