> [!NOTE] Lemma (Stable Stationary Point) > Let $y_{k}=u(y_{k-1}), \quad \text{for all }k\geq 1$where $u:\mathbb{R}\to \mathbb{R}$ is a given function and $y_{0}=\bar{y}$, be a [[First Order Autonomous Recurrence Relation|first order autonomous recurrence relation]]. Let $y^{*}$ be a [[Stationary Point of First Order Autonomous Recurrence Relation|stationary point]] of the equation. Then $y^{*}$ is [[Stable Stationary Point of First Order Autonomous Recurrence Relation|stable]] if $|u'(y^{*})|<1$ and unstable if $|u'(y^{*})|>1.$ **Informal justification**: Note that $y_{1}=u(y_{0})\approx u(y^{*})+u'(y^{*})(y_{0}-y^{*}) =y^{*}+u'(y^{*})(y_{0}-y^{*})$Thus $|y_{1}-y^{*}|\lessapprox |u'(y^{*})||y_{0}-y^{*}|.$We see that, approximately, the distance decreases if $|u'(y^{*})|<1.$