\(\frac{a}{b}< \frac{c}{d}\Rightarrow ad< bc\)
Có:
\(\frac{ab+ad}{b\left(b+d\right)}< \frac{ab+bc}{b\left(b+d\right)}\)\(\Rightarrow\frac{a\left(b+d\right)}{b\left(b+d\right)}< \frac{b\left(a+c\right)}{b\left(b+d\right)}\)
\(\Rightarrow\frac{a}{b}< \frac{a+c}{b+d}\left(1\right)\)
\(\frac{ad+cd}{d\left(b+d\right)}< \frac{bc+cd}{d\left(b+d\right)}\)\(\Rightarrow\frac{d\left(a+c\right)}{d\left(b+d\right)}< \frac{c\left(b+d\right)}{d\left(b+d\right)}\)
\(\Rightarrow\frac{a+c}{b+d}< \frac{c}{d}\left(2\right)\)
\(\left(1\right)\left(2\right)\Rightarrow\frac{a}{b}< \frac{a+c}{b+d}< \frac{c}{d}\)