Editing Limit of a sequence (section)

===Iterated limit===

For a double sequence <math>(x_{n,m})</math>, we may take limit in one of the indices, say, <math>n \to \infty</math>, to obtain a single sequence <math>(y_m)</math>, and then take limit in the other index, namely <math>m \to \infty</math>, to get a number <math>y</math>. Symbolically,
:<math>\lim_{m \to \infty} \lim_{n \to \infty} x_{n, m} = \lim_{m \to \infty} y_m = y</math>.

This limit is known as '''[[iterated limit]]''' of the double sequence. The order of taking limits may affect the result, i.e.,

:<math>\lim_{m \to \infty} \lim_{n \to \infty} x_{n, m} \ne \lim_{n \to \infty} \lim_{m \to \infty} x_{n, m}</math> in general.

A sufficient condition of equality is given by the [[Moore-Osgood theorem]], which requires the limit <math>\lim_{n \to \infty}x_{n, m} = y_m</math> to be uniform in <math display="inline">m</math>.<ref name="Zakon" />