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Suppose that $a$, $b$ and $c$ are positive real numbers and that $a \leq b + c$. By cross multiplying or otherwise, show that

$$\frac{a}{1 + a} \leq \frac{b}{1 + b} + \frac{c}{1 + c}$$

Anyone able to get this to work out? I cross multiplied them but am unable to manipulate it into an expression that verifies the statement.

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By cross multipliying since $a,b>0$ \begin{equation}a(1+b)(1+c)\le b(1+a)(1+c)+c(1+a)(1+b)\Leftrightarrow a+ab+ac+abc\le b+ba+bc+bac+c+ca+cb+cab\Leftrightarrow a\le b+c+2bc+abc \end{equation} which holds since $a\le b+c$ and $a,b,c>0$

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