Factorizations controlling movement around the positive fixed point

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The first difference shows whether an update raises or lowers the current state; the second shows whether it lies above or below one. For 1/5≤z<1, both expressions imply z<F(z)≤1. These sign relations support a monotone convergence argument within that invariant positive interval.

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Original context

6.3 Proposition 3: every positive averaged trajectory converges to one

Two factorizations establish the result:

Fz−z=1−z2z+17z,  Fz−1=5z−1z−17z.

For z>1, these show 1<Fz<z. The sequence decreases while bounded below by one. For 1/5≤z<1, they show z<Fz≤1, so the sequence increases while bounded above by one. For 0<z<1/5, the next value is greater than one and thereafter belongs to the first case. A bounded monotone sequence has a limit; continuity and the positive fixed-point equation force that limit to be one.

For z0=2, local computation gives:

Iteration

F\left(z\right)-z=\frac{\left(1-z\right)\left(2z+1\right)}{7z},  F\left(z\right)-1=\frac{\left(5z-1\right)\left(z-1\right)}{7z}.