<s>[INST] Given positive real numbers $a, b, c$ with $aI am trying to prove the following:
$$\frac{a}{1+a} < \frac{b}{1+b} + \frac{c}{1+c}$$
given that $a, b, c > 0$ and $a < b+c$. I tried various rearrangements but can't seem to get anywhere with it.
 [/INST] $$\frac{b}{1+b} + \frac{c}{1+c}>\frac{a}{a+1}$$
$$\frac{b+c+2bc}{1+b+c+bc}>\frac{a}{a+1}$$
Since $a,b,c>0$, 
$$ab+ac+2abc+b+c+2bc>a+ab+ac+abc$$
$$abc+2bc+b+c>a$$
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