→Proof: Matched math sequence bracket notation with that of math sequence article, and corrected the inconsistent use of variable name x outside math mode
<blockquote>'''Banach Fixed Point Theorem.''' Let <math>(X, d)</math> be a [[Empty set|non-empty]] [[complete metric space]] with a contraction mapping <math>T : X \to X.</math> Then ''T'' admits a unique [[Fixed point (mathematics)|fixed-point]] ''<math>x^*''</math> in ''X'' (i.e. ''<math>T''(''x^*'') = ''x^*'')</math>. Furthermore, ''<math>x^*''</math> can be found as follows: start with an arbitrary element <math>x_0 \in X</math> and define a [[sequence]] {''x<submath>(x_n)_{n\in\mathbb N}</submath>''} by ''x<submath>n</sub>''x_n = ''T''(''x''<sub>''x_{n''−1-1})</submath>) for <math>n \geq 1.</math> Then {{nowrap|''x<submath>\lim_{n</sub>''→\to \infty} x_n = ''x^*''}}</math>.</blockquote>
''Remark 1.'' The following inequalities are equivalent and describe the [[Rate of convergence|speed of convergence]]:
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</math>
Any such value of ''q'' is called a ''[[Lipschitz constant]]'' for ''<math>T''</math>, and the smallest one is sometimes called "the best Lipschitz constant" of ''<math>T''</math>.
''Remark 2.'' ''<math>d''(''T''(''x''), ''T''(''y'')) < ''d''(''x'', ''y'')</math> for all ''<math>x''≠\neq ''y''</math> is in general not enough to ensure the existence of a fixed point, as is shown by the map '' :<math>T'' : [1, ∞\infty) →\to [1, ∞\infty), ''\,\, T''(''x'') = ''x'' + \tfrac{1/''}{x''}\,,</math> which lacks a fixed point. However, if ''<math>X''</math> is [[Compact space|compact]], then this weaker assumption does imply the existence and uniqueness of a fixed point, that can be easily found as a minimizer of ''<math>d''(''x'', ''T''(''x''))</math>, indeed, a minimizer exists by compactness, and has to be a fixed point of ''<math>T''</math>. It then easily follows that the fixed point is the limit of any sequence of iterations of ''<math>T''</math>.
''Remark 3.'' When using the theorem in practice, the most difficult part is typically to define ''<math>X''</math> properly so that <math>T(X) \subseteq X.</math>
