$S = \dfrac{1}{2^{2}} + \dfrac{1}{3^{2}} + + \dfrac{1}{4^{2}} + ... + \dfrac{1}{2009^{2}}$
$\to S < \dfrac{1}{1.2} + \dfrac{1}{2.3} + \dfrac{1}{3.4} + ... + \dfrac{1}{2008.2009}$
$\to S < 1- \dfrac{1}{2} + \dfrac{1}{2} - \dfrac{1}{3} + \dfrac{1}{3} - \dfrac{1}{4} + ... + \dfrac{1}{2008} - \dfrac{1}{2009}$
$\to S < 1 - \dfrac{1}{2009} < 1$