Ta có :
$A = \dfrac{1}{1.2}+\dfrac{1}{2.3}+...+\dfrac{1}{99.100} $
$=\dfrac{2-1}{1.2}+\dfrac{3-2}{2.3}+...+\dfrac{100-99}{99.100} $
$=\dfraac{1}{1}-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{99}-\dfrac{1}{100} $
$=1-\dfrac{1}{100} < 1$
Vậy $A <1$