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In a lecture I once attended, I remember the speaker using a result of the following nature: $``$Let $\{A_n\}_{n=1}^\infty \subset \mathbb R$ be a sequence satisfying a recursion of the form $$P(n) ...

Let $a_1=0$ and let $ - \ln(2) < a_2 < \ln(2) $ Define $$ a_n = a_{n-1}^3 - a_{n-2} $$ Then $$ \sup_{n>2} a_n = a_2 $$ And $$ \inf_{n>2} a_n = - a_2 $$ How to prove that ?

Fix $0 < c < 1/3$. Show that there exists a strictly increasing sequence $a_k > 0$, a sequence $b_k \geq 0$, and an infinite set $K \subseteq \mathbb{Z}_{>0}$ such that \begin{align}\...

I have the following recurrent relation and I want to find a close form of it if it exists at all. $$ P_n = (1-p)^{n-1}P_{n-1} + \sum\limits_{k=2}^{n} \binom{n-1}{k-1} p^{\binom{k}{2}} (1-p)^{k(n-k)} ...