I have a sequence defined using the following recursion formula:
$M_0 = N = 1000$
$M_{t+1} = \frac {M_t} {1 + \left( \frac1{e-1} + \frac{\sqrt{N/M_t}-1}{\sqrt{2}} \right)^{-1}}$
I would like to know its limit when $t -> \inf$.
I note that $M_{t+1} <= M_t$, so the best candidate to the limit is 0.
However, when $M_t$ goes to 0, $M_{t+1} / M_t$ goes to 1. So, I don't know if and how I can prove that the limit is 0?