IrMO 1991 Problem 7

Let $$a_n=\frac{n^2+1}{\sqrt{n^4+4}}, n=1,2,3,\dots$$ and let $b_n$ be the product of $a_1,a_2,a_3,\dots ,a_n$ . Prove that $$\frac{b_n}{\sqrt{2}}=\frac{\sqrt{n^2+1}}{\sqrt{n^2+2n+2}},$$ and deduce that $$\frac{1}{n^3+1}<\frac{b_n}{\sqrt{2}}-\frac{n}{n+1}<\frac{1}{n^3}$$ for all positive integers $n$ .