Problem 35 (difficulty: 4/10)

 (1)  Let \(\displaystyle a_1 =1\) and \(\displaystyle a_{n+1} =\sqrt{2a_n +3}\). Prove that \(\displaystyle \forall n\in \N \ a_n \le a_{n+1}\).
 (2)  Let \(\displaystyle a_1 =0.9\) and \(\displaystyle a_{n+1} =a_n -a_n^2 \). Prove that \(\displaystyle \forall n\in \N\ a_{n+1} <a_n \;\) and \(\displaystyle \;0<a_n <1\).

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