If $a_n$ is a decreasing sequence of real numbers and $a_0=1$. How to prove the minimum value of $\sum a_n^2/a_{n+1}$ is 4 without using Cauchy Schwarz inequality? Here's what I got:
If $a_n=1/2^n$, then $a_n^2/a_{n+1}>1/2^n$. I try to prove the minimum is attained when sequence $a_n=1/2^n$ using induction but I failed.