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Sorry if I misuse or soil on math's syntax and formalism, but how do you show that a recursive function converges? Or has a fixed point?

For example, in $x_{n+1}=x_n^2-2$, for an initial value of $-2<x_0>2$, it becomes increasingly large. For values of $x_0=0,-2,-1,1,2$, it collapses to a fixed point. For values $-2<x_0<2$, except those mention earlier, the function does not seem to converge. How so?

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It is because you can study $x \mapsto x^2-2$ and find a steady interval in which you will have monotonicity.

In general case, when you have a sequence defined as $u_{n+1}=f\left(u_n\right)$ you should study variations of $f$. If it increases on a steady interval $I$, the sequence with $u_0 \in I$ will be monotone, if decreasing you need to study $u_{2n}$ and $u_{2n+1}$.

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