As part of an exercise I am given a sequence defined by $a_1 = 1$
and $$a_{n+1} = 1 + \frac{1}{1 + a_n}$$
I have noticed that the even sequence is decreasing and I want to prove this, the even sequence (and even the odd) will be given by
$$a_{n+2} = \frac{4 + 3a_n}{3 + 2a_n}$$
I proceed by induction and check the base case, then I suppose $a_{n+2} < a_{n}$ to prove that $a_{n+4} < a_{n+2}$. But through numerous substitutions I arrive nowhere.
I have the habit of posting my calculations but I lead myself only to dead ends, is induction the wrong way to go in this case? Could I bother you with a proof?