MathNet 2008-part-03 — 165 problems (EN–VI) Question 5

Prove that if $\left|\frac{a+b}{2}\right| + \left|\frac{a-b}{2}\right| \lt c$, for $a, b, c \in \mathbb{R}$, then $|a| \lt c$ and $|b| \lt c$.