Editing Complete metric space (section)

==Definition==

'''Cauchy sequence'''

A [[sequence]] <math>x_1, x_2, x_3, \ldots</math> of elements from <math>X</math> of a [[metric space]] <math>(X, d)</math> is called '''Cauchy''' if for every positive [[real number]] <math>r > 0</math> there is a positive [[integer]] <math>N</math> such that for all positive integers <math>m, n > N,</math> <math display=block>d(x_m, x_n) < r.</math> 

'''Complete space'''

A metric space <math>(X, d)</math> is '''complete''' if any of the following equivalent conditions are satisfied:
#Every Cauchy sequence in <math>X</math> converges in <math>X</math> (that is, has a limit that is also in <math>X</math>).
#Every decreasing sequence of [[empty set|non-empty]] [[closed subset]]s of <math>X,</math> with [[Diameter of a set|diameters]] tending to&nbsp;0, has a non-empty [[Intersection (set theory)|intersection]]: if <math>F_n</math> is closed and non-empty, <math>F_{n+1} \subseteq F_n</math> for every <math>n,</math> and <math>\operatorname{diam}\left(F_n\right) \to 0,</math> then there is a unique point <math>x \in X</math> common to all sets <math>F_n.</math>