$\displaystyle if \ \frac {a}{b} < \frac{c}{d} \ show \ that \ : \ \frac{a}{b}<\frac{a+c}{b+d}<\frac{c}{d} .... b,d>0 $
Here is one-half of it.
$\displaystyle \begin{gathered}
\frac{a}
{b} < \frac{c}
{d} \hfill \\
ad < bc \hfill \\
ab + ad < ab + bc \hfill \\
a(b + d) < b(a + c) \hfill \\
\frac{a}
{b} < \frac{{a + c}}
{{b + d}} \hfill \\
\end{gathered} $