Stable affine recurrence toward one

Read the equation

Subtracting one from both sides gives error_next=a×error. For 0<a<1, the positive error shrinks geometrically but remains nonzero at every finite step when initialized above one. This supplies an ordinary nonconscious dynamical example of persistent finite-time discrepancy with eventual convergence.

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Original context

7.4 Proposition 6: elementary controllers satisfy permanent origin divergence

For any a with 0<a<1, let

zn+1=azn+1−a,  z0=2.

The exact solution is

zn=1+an.

For every integer n≥1, 1<zn<2. The trajectory never returns exactly to the origin two, although it converges to one. A routine can repeatedly output zn=zn at every step. This satisfies the proposed weak numerical pattern of an internally computed change, permanent origin difference, and repeated declarations.

{z}_{n+1}=a{z}_{n}+\left(1-a\right),  {z}_{0}=2.