Ta có:
$\dfrac{1}{1.2} + \dfrac{1}{2.3} + \dfrac{1}{3.4} + .... + \dfrac{1}{99.100}$
$= 1 - \dfrac{1}{2} + \dfrac{1}{2} - \dfrac{1}{3} + \dfrac{1}{3} - \dfrac{1}{4} + .... + \dfrac{1}{99} - \dfrac{1}{100}$
$= 1 - \dfrac{1}{100}$
Mà $1 - \dfrac{1}{100} <1$ $⇒$ $\dfrac{1}{1.2} + \dfrac{1}{2.3} + \dfrac{1}{3.4} + .... + \dfrac{1}{99.100} < 1$ ($đpcm$)