Let there is $\lambda<5$ such that the inequality $0 <\sqrt {2002} - \frac {a} {b} <\frac {\lambda} {ab}$ holds for an infinite number of pairs $ (a, b)$ of positive integers or $a<\sqrt{2002}b<a+\frac{\lambda}{a}$ $a^2<2002b^2<a^2+\frac{\lambda^2}{a^2}+2\lambda<a^2+10$ for big enough $a$ so $1\leq 2002b^2-a^2 \leq 9$ for an infinite number of pairs $ (a, b)$ Let $a^2=2002b^2-t, 1 \leq t \leq 9$ $a^2\equiv 1,3,4,5,9 \pmod{11} \to t\equiv 2,6,7,8,10 \pmod{11} \to t=2,6,7,8$ $a^2\equiv 1,3,4,9,10,12 \pmod{13} \to t\equiv 1,3,4,9,10,12 \pmod{13} \to t=1,3,4,9$ -contradiction. So $ \lambda \geq 5$