Problem

Source: RMO 2018 P6

Tags: algebra



Define a sequence $\{a_n\}_{n\geq 1}$ of real numbers by \[a_1=2,\qquad a_{n+1} = \frac{a_n^2+1}{2}, \text{ for } n\geq 1.\]Prove that \[\sum_{j=1}^{N} \frac{1}{a_j + 1} < 1\]for every natural number $N$.