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The smallest such value of ''q'' is sometimes called the ''[[Lipschitz constant]]''.
Note that the requirement d(''Tx'', ''Ty'') < d(''x'', ''y'') for all unequal ''x'' and ''y'' is in general not enough to ensure the existence of a fixed point, as is shown by the map ''T'' : <nowiki>[1,∞) → [1,∞)</nowiki> with ''T''(''x'') = ''x'' + 1/''x'', which lacks a fixed point. However, if the space ''X'' is [[Compact space|compact]], then this weaker assumption does imply all the statements of the theorem.
When using the theorem in practice, the most difficult part is typically to define ''X'' properly so that ''T'' actually maps elements from ''X'' to ''X'', i.e. that ''Tx'' is always an element of ''X''.
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