Problem

Source: China Sijiazhuang , Aug 2014

Tags: inequalities proposed, inequalities, algebra, China



Define a positive number sequence sequence $\{a_n\}$ by \[a_{1}=1,(n^2+1)a^2_{n-1}=(n-1)^2a^2_{n}.\]Prove that\[\frac{1}{a^2_1}+\frac{1}{a^2_2}+\cdots +\frac{1}{a^2_n}\le 1+\sqrt{1-\frac{1}{a^2_n}} .\]