Let $f:\mathbb{R} \rightarrow (0,1)$ be any function (possibly not continous) whose ranges lies in $(0,1)$. For a recursive sequence $a_1=1, a_2=f(a_1), a_3=\frac{1}{a_2},a_4=f(a_3),...,a_n=1/a_{n-1} $ if n is odd or $a_n=f(a_{n-1})$ if n is even.
How do I prove that $a_n$ has a convergent subsequence? I have a strong feeling that I should argue that the sequence is bounded, and use the Bolzano-Weierstrass Theorem to prove that a convergent subsequence exists. However, I don't see any bounds for the sequence...