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 function f on the
interval [a,b] if and only if
limn→∞sup{|fn(x)-f(x)|,a≤x≤b}=0. |
Proof.
Suppose the sequence converges uniformly. By the very definition of
uniform convergence, we have that for any ϵ there exist N
such that
|fn(x)-f(x)|<ϵ2,a≤x≤b |
hence
Conversely, suppose the sequence does not converge uniformly. This means that there is an for which there is a sequence of increasing integers and points with the corresponding subsequence of functions such that
therefore
Consequently, it is not the case that
∎
Theorem 2.
The uniform limit of a sequence of continuous complex or real
functions in the interval is continuous in
The proof is here (http://planetmath.org/LimitOfAUniformlyConvergentSequenceOfContinuousFunctionsIsContinuous)
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 |