# Math Help - inequality and fractions

1. ## inequality and fractions

I have been working on this for a little bit now, but I can't get it.
$\mbox{If } a \leq b + c \mbox{ and } a,b,c \geq 0, \mbox{ show } \frac{a}{a+1} \leq \frac{b}{b+1} + \frac{c}{c+1}$. Its the very first question in a series of questions on metric spaces, so I have a feeling I am missing something obvious. Thank you.

2. $\color{white}.\quad.$ $a\ \leqslant\ b+c\ \leqslant\ abc+2bc+b+c$

$\Rightarrow\ a-abc+1-bc\ \leqslant\ bc+b+c+1$

$\Rightarrow\ (a+1)(1-bc)\ \leqslant\ (b+1)(c+1)$

$\Rightarrow\ \frac{1-bc}{(b+1)(c+1)}\ \leqslant\ \frac1{a+1}$

$\Rightarrow\ \frac1{b+1}+\frac1{c+1}-1\ \leqslant\ \frac1{a+1}$

$\Rightarrow\ 1-\frac1{a+1}\ \leqslant\ 1-\frac1{b+1}\,+\,1-\frac1{c+1}$

$\Rightarrow\ \frac{a}{a+1}\ \leqslant\ \frac{b}{b+1}+\frac{c}{c+1}$