RMO 2018 K6

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