I am wanting to prove that the following recursive sequence is monotonic decreasing via proof by induction.
Let $ S_1 = 1, ~ S_{n+1} = \frac{n}{n+1} (S_n)^2;~ n \geq 1. $
Here is what I have so far but I feel the proof fails at my last statement and I am unsure how to correct it.
$$ \text{Base:} ~ S_1 = 1 > \frac{1}{2} = S_2 \\ $$ $$ \text{Assumption:} ~ S_{k+1} > S_{k+2} \\ $$ $$ \text{Proof for:} ~ k+2: $$ $$ \text{ie:} ~ S_{k+1} > S_{k+2} \\ \Rightarrow S_{k+2} = \frac{k+1}{k+2} (S_{k+1})^2 \\ \Rightarrow S_{k+2} =\frac{k+1}{k+2}(\frac{k}{k+1})^2S_{k}^4 \\ \Rightarrow S_{k+2} = \frac{k^2}{(k+1)(k+2)}S_{k}^4 < \frac{k^2}{(k+1)(k+1)}S_{k}^4 = [(\frac{k}{k+1})S_k^2]^2 = S_{k+1}^2 \\ \text{Since}~ S_{k+1}^2 > S_{k+2} \Rightarrow S_{k+1} > S_{k+2} $$
Is this fine or have I messed up badly?
Any help/hints is/are appreciated.
Best Answer
HINT: Use the fact that $x^2<x$ whenever $0<x<1$. I’ve left the details spoiler-protected.