Ta có \(a,b,c\) dương
\(\Leftrightarrow\left\{{}\begin{matrix}a+b>0\\b+c>0\\c+a>0\end{matrix}\right.\)
Ta có :
\(\dfrac{a}{a+b+c}< \dfrac{a}{a+b}< \dfrac{a+c}{a+b+c}\left(1\right)\)
\(\dfrac{b}{a+b+c}< \dfrac{b}{b+c}< \dfrac{a+b}{a+b+c}\) \(\left(2\right)\)
\(\dfrac{c}{a+b+c}< \dfrac{c}{a+c}< \dfrac{b+c}{a+b+c}\left(3\right)\)
Cộng từng vế của \(\left(1\right),\left(2\right),\left(3\right)\) ta được :
\(\dfrac{a}{a+b+c}+\dfrac{b}{a+b+c}+\dfrac{c}{a+b+c}< \dfrac{a}{a+b}+\dfrac{b}{b+c}+\dfrac{c}{c+a}< \dfrac{a+c}{a+b+c}+\dfrac{a+b}{a+b+c}+\dfrac{b+c}{a+b+c}\)
\(\Leftrightarrow\dfrac{a+b+c}{a+b+c}< \dfrac{a}{a+b}+\dfrac{b}{b+c}+\dfrac{c}{c+a}< \dfrac{2\left(a+b+c\right)}{a+b+c}\)
\(\Leftrightarrow1< \dfrac{a}{a+b}+\dfrac{b}{b+c}+\dfrac{c}{c+a}< 2\)
\(\Leftrightarrow1< M< 2\)
\(\Leftrightarrow M\notin Z\left(đpcm\right)\)
