Limes superior and inferior inequalities with functions and its derivatives Lemma 2. from N.H. Duu, On The Existence of Bounded Solutions for Lotka- Volterra Equations, Acta Mathematica Vietnamica 25(2) (2000), 145-159.

Let $G(t)$ and $F(t)$ be two diferentiable functions defined on
  $(0,\infty)$ such that $\lim_{t\to\infty} G(t) = \lim_{t\to\infty}F(t)
 = +\infty$ then $$\limsup_{t\to\infty}\frac{G(t)}{F(t)}\leq \limsup_{t\to\infty}\frac{G'(t)}{F'(t)};
 \quad\liminf_{t\to\infty}\frac{G(t)}{F(t)}\geq
 \liminf_{t\to\infty}\frac{G'(t)}{F'(t)}.$$
Proof. By the Cauchy theorem for diferential functions, for any
  $t_{1}, t_{2} > 0$ there is an $\theta \in (t_{1}, t_{2})$ such that
  $$\frac{G'(\theta)}{F'(\theta)}=\frac{G(t_{1})-G(t_{2})}{F(t_{1})-F(t_{2})}=\dfrac{G(t_{2})}{F(t_{2})}\times\dfrac{1-\dfrac{G(t_{1})}{G(t_{2})}}{1-\dfrac{F(t_{1})}{F(t_{2})}}.$$
  Letting $t_{1}$ and $t_{2} \to \infty$ such that $\lim\dfrac{G(t_{1})}{G(t_{2})}
 = lim \dfrac{F(t_{1})}{F(t_{2})}= 0$ we get the result.

Can anybody help me understand the last part of this proof? Exactly, how this transformation $$\frac{G'(\theta)}{F'(\theta)}=\frac{G(t_{1})-G(t_{2})}{F(t_{1})-F(t_{2})}=\dfrac{G(t_{2})}{F(t_{2})}\times\dfrac{1-\dfrac{G(t_{1})}{G(t_{2})}}{1-\dfrac{F(t_{1})}{F(t_{2})}} \text{where} \lim_{t_1,t_2\to\infty}\dfrac{G(t_{1})}{G(t_{2})}
 = \lim_{t_1,t_2\to\infty} \dfrac{F(t_{1})}{F(t_{2})}= 0$$ leads to inequalities from lemma? 
 A: The cited proof lacks many details. So I expand the limit superior version. The inferior version is analogous. 


*

*Since $F(t) \to \infty$ for $t\to\infty$ we may assume that $F > 0$ without lost of generality. 

*Fix some $t_1\in (0,\infty)$. We may assume that for every $t > t_1$ it follows $F(t) > F(t_1)$.

*By Cauchy mean value theorem, for each $t_2 > t_1$ there exists some $\theta \in (t_1, t_2)$ with
$$ 
\frac{G(t_2) - G(t_1)}{F(t_2)} 
\le \frac{G(t_2) - G(t_1)}{F(t_2) - F(t_1)} 
= \frac{G'(\theta)}{F'(\theta)}
\le \sup_{t > t_1} \frac{G'(t)}{F'(t)}
=: M(t_1).
$$

*In particular for $t_2 > t_1$, we have
$$ 
\frac{G(t_2)}{F(t_2)}
= \frac{G(t_2) - G(t_1)}{F(t_2)} + \frac{G(t_1)}{F(t_2)}
\le M(t_1) + \frac{G(t_1)}{F(t_2)}.
$$

*Compute the limit superior with respect to $t _2 $:
$$
\limsup_{t_2\to\infty} \frac{G(t _2)}{F(t _2)} \le M(t_1) + 0.
$$
Notice that the left-hand-side becomes independent of $t_1$. 

*Now apply the limit with respect to $t_1$:
$$
\limsup_{t _2\to\infty} \frac{G(t _2)}{F(t _2)} \le \lim_{t_1\to \infty} M(t_1) 
= \limsup_{t\to\infty} \frac{G'(t)}{F'(t)}.
$$
And we are done. 
