How could I prove the following statement without using induction? I've been staring at this for the better part of an hour. (To be fair, I'm not very good at proof writing) Thanks in advance!
Define a sequence $a_n, n \ge 0,$ inductively by $a_0 = 2,$ and for all $n \ge 0, a_{n+1} = \sqrt{a_n + 1}.$
Using the fact that the polynomial $x^2 - x - 1 < 0$ if and only if $\frac{1-\sqrt{5}}{2} < x < \frac{1+\sqrt{5}}{2}$, prove that for every $n \ge 0, a_n > a_{n+1}.$