FI-MO 2001 Problem 3

Numbers $a, b$ and $c$ are positive integers and $\frac{1}{a}+\frac{1}{b}+\frac{ 1}{c}< 1.$ Show that $$\frac{1}{a}+\frac{1}{b}+\frac{ 1}{c}\leq \frac{41}{42}.$$