condition for uniform convergence of sequence of functions

Proof of limits of functionsFernando Sanz Gamiz

Theorem 1.

Let  f1,f2,  be a sequence of real or complex functions defined on the interval[a,b].  The sequence converges uniformly to the limit functionMathworldPlanetmath f on the interval  [a,b] if and only if



Suppose the sequence converges uniformly. By the very definition of uniform convergenceMathworldPlanetmath, we have that for any ϵ there exist N such that

|fn(x)-f(x)|<ϵ2,axb   for n>N


sup{|fn(x)-f(x)|,axb}<ϵ   for n>N

Conversely, suppose the sequence does not converge uniformly. This means that there is an ϵ for which there is a sequence of increasing integers ni,i=1,2, and points xni with the corresponding subsequence of functions fni such that

|f(xni)-fni(xni)|>ϵ  for all i=1,2,


sup{|fn(x)-f(x)|,axb}>ϵ   for infinitely many n.

Consequently, it is not the case that


Theorem 2.

The uniform limit of a sequence of continuousMathworldPlanetmath complex or real functions fn in the interval [a,b] is continuous in [a,b]

The proof is here (

Title condition for uniform convergence of sequence of functions
Canonical name ConditionForUniformConvergenceOfSequenceOfFunctions
Date of creation 2013-03-22 17:07:49
Last modified on 2013-03-22 17:07:49
Owner fernsanz (8869)
Last modified by fernsanz (8869)
Numerical id 6
Author fernsanz (8869)
Entry type Proof
Classification msc 40A30
Classification msc 26A15