Banach fixed-point theorem: Difference between revisions

Revision as of 11:20, 21 December 2023

Revision as of 11:24, 21 December 2023

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

This follows by [[Principle of mathematical induction|induction]] on ''n'', using the fact that ''T'' is a contraction mapping. Then we can show that <math>(x_n)_{n\in\mathbb N}</math> is a [[Cauchy sequence]]. In particular, let <math>m, n \in \N</math> such that ''<math>m'' > ''n'' </math>:

: <math>\begin{align}

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\end{align}</math>

Let ε > 0 be arbitrary. Since ''<math>q'' ∈\in [0, 1)</math>, we can find a large <math>N \in \N</math> so that

:<math>q^N < \frac{\varepsilon(1-q)}{d(x_1, x_0)}.</math>

Therefore, by choosing ''<math>m''</math> and ''<math>n''</math> greater than ''<math>N''</math> we may write:

:<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>