Banach fixed-point theorem: Difference between revisions

Revision as of 07:18, 16 September 2024

Revision as of 07:21, 16 September 2024

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:<math>d(x_m, x_n) \leq q^n d(x_1, x_0) \left ( \frac{1}{1-q} \right ) < \left (\frac{\varepsilon(1-q)}{d(x_1, x_0)} \right ) d(x_1, x_0) \left ( \frac{1}{1-q} \right ) = \varepsilon.</math>

This proves that the sequence <math>(x_n)_{n\in\mathbb N}</math> is Cauchy. By completeness of <math>(''X'','' d'')</math>, the sequence has a limit <math>x^* \in X.</math> Furthermore, <math>x^*</math> must be a [[Fixed point (mathematics)|fixed point]] of <math>T</math>:

:<math>x^*=\lim_{n\to\infty} x_n = \lim_{n\to\infty} T(x_{n-1}) = T\left(\lim_{n\to\infty} x_{n-1} \right) = T(x^*). </math>