Let $N \in \mathbb{N}$ where $N \geq 3$, $r \in \left(\frac{2N}{N+2},2\right]$ and $p-1 \in \left(1,\frac{N+2}{N-2}\right)$. Define for each $j \in \mathbb{N}$ recursively the expressions: $$\frac{1}{t_1}=\frac{1}{r}-\frac{2}{N}>0,$$ where the expression above is positive by assumption, and $$\frac{1}{t_j}=\frac{p-1}{t_{j-1}}-\frac{2}{N}. $$
My question is: There exist some $j_0 \in \mathbb{N}$ such that $$\frac{p-1}{t_{j_0}}-\frac{2}{N} \leq 0\,\,?$$
I've tried to find some pattern for $t_j$, but the expressions get very complex very fast. Any hint on how to proceed to decide if such $j_0$ exists will be very helpful, thank you.
For the context of the problem, this question arised when deciding if an bootstrap argument for the regularization of an elliptic PDE is a finite process.
$\left(\frac1{t_j}\right)$ follows a (first order) linear recurrence with constant coefficients hence you may easily find its closed form and deduce its sign.