Proof of an algebraic inequality

Question

Show that for $A,B,C \in \mathbb{R}$ and $A,B,C \ge 0$ that $A \le B + C \implies \frac{A}{1+A} \le \frac{B}{1+B} + \frac{C}{1+C}$

Answer 1

Observe $\dfrac{B}{1+B} + \dfrac{C}{1+C} \ge \dfrac{B}{1+B+C}+\dfrac{C}{1+B+C} = \dfrac{B+C}{1+B+C} \ge \dfrac{A}{1+A}$. This last one is true since $B+C \ge A$.