Editing Sequence (section)

=== Formal definition of convergence===

A sequence of real numbers <math>(a_n)</math> '''converges to''' a real number <math>L</math> if, for all <math>\varepsilon > 0</math>, there exists a natural number <math>N</math> such that for all <math>n \geq N</math> we have<ref name="Gaughan">{{cite book|title=Introduction to Analysis |last=Gaughan |first=Edward |publisher=AMS (2009)|isbn=978-0-8218-4787-9|chapter=1.1 Sequences and Convergence|year=2009 }}</ref> 
:<math>|a_n - L| < \varepsilon.</math>

If <math>(a_n)</math> is a sequence of complex numbers rather than a sequence of real numbers, this last formula can still be used to define convergence, with the provision that <math>|\cdot|</math> denotes the complex modulus, i.e. <math>|z| = \sqrt{z^*z}</math>. If <math>(a_n)</math> is a sequence of points in a [[metric space]], then the formula can be used to define convergence, if the expression <math>|a_n-L|</math> is replaced by the expression <math>\operatorname{dist}(a_n, L)</math>, which denotes the [[Metric (mathematics)|distance]] between <math>a_n</math> and <math>L</math>.