Editing Uniform convergence (section)

=== Notes ===

Intuitively, a sequence of functions <math>f_n</math> converges uniformly to <math>f</math> if, given an arbitrarily small <math>\varepsilon>0</math>, we can find an <math>N\in\N</math> so that the functions <math>f_n</math> with <math>n>N</math> all fall within a "tube" of width <math>2\varepsilon</math> centered around <math>f</math> (i.e., between <math>f(x)-\varepsilon</math> and <math>f(x)+\varepsilon</math>) for the ''entire domain'' of the function.

Note that interchanging the order of quantifiers in the definition of uniform convergence by moving "for all <math>x\in E</math>" in front of "there exists a natural number <math>N</math>" results in a definition of [[pointwise convergence]] of the sequence. To make this difference explicit, in the case of uniform convergence, <math>N=N(\varepsilon)</math> can only depend on <math>\varepsilon</math>, and the choice of <math>N</math> has to work for all <math>x\in E</math>, for a specific value of <math>\varepsilon</math> that is given. In contrast, in the case of pointwise convergence, <math>N=N(\varepsilon,x)</math> may depend on both <math>\varepsilon</math> and <math>x</math>, and the choice of <math>N</math> only has to work for the specific values of <math>\varepsilon</math> and <math>x</math> that are given. Thus uniform convergence implies pointwise convergence, however the converse is not true, as the example in the section below illustrates.