Banach fixed-point theorem: Difference between revisions

Revision as of 19:37, 20 November 2023

Revision as of 11:20, 21 December 2023

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==Proof==

Let <math>x_0 \in X</math> be arbitrary and define a [[sequence]] <math>(x_n)_{n\in\mathbb N}</math> by setting ''x<submath>n</sub>''x_n = ''T''(''x''<sub>''x_{n''−1-1})</submath>). We first note that for all <math>n \in \N,</math> we have the inequality

:<math>d(x_{n+1}, x_n) \le q^n d(x_1, x_0).</math>