<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] Function $f(x)=\frac{x}{1+x}$ is increasing (it is easy to check first derivative). Therefore $f(a)<f(b+c)$, for $a<b+c$. Because of that, we have $$\frac{a}{1+a}<\frac{b+c}{1+b+c}=\frac{b}{1+b+c}+\frac{c}{1+b+c},$$ from where we get (since $a,b,c>0$):
$$\frac{a}{1+a}<\frac{b}{1+b}+\frac{c}{1+c}.$$
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