Mediant Inequality Proof: $\frac{a}{b} < \frac{a+c}{b+d} <\frac{c}{d}$ If $\frac{a}{b}$ < $\frac{c}{d}$ then 
$$\frac{a}{b} < \frac{a+c}{b+d} <\frac{c}{d}$$ 
I have been searching and trying and couldnt find a reliable proof. 
One might do this like here which I think it is very wrong:
$\frac{a}{b}$ < $\frac{c}{d}$ then $a.d$ < $c.b$
Because $\frac{2}{-3}$ < $\frac{4}{5}$ doesn't mean $10 < -12$

Hint. As regards $\frac{a}{b} < \frac{a+c}{b+d}$, show that
$$\frac{a+c}{b+d}-\frac{a}{b}=\frac{\frac{c}{d}-\frac{a}{b}}{\frac{b}{d}+1}.$$
