CWMO 2004 Problem 8

Suppose that $a$ , $b$ , $c$ are positive real numbers, prove that $$1 < \frac {a}{\sqrt {a^{2} + b^{2}}} + \frac {b}{\sqrt {b^{2} + c^{2}}} + \frac {c}{\sqrt {c^{2} + a^{2}}}\leq\frac {3\sqrt {2}}{2}$$