$A=1+\dfrac{1}{2^2} +\dfrac{1}{3^2} +\dfrac{1}{4^2}...+\dfrac{1}{n^2}< 1+\dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+...+\dfrac{1}{(n-1)n}$
$=1+\bigg(1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4} +...+\dfrac{1}{n-1} -\dfrac{1}{n} \bigg)=2-\dfrac{1}{n}$