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Uniform convergence #definition

{fn(x)} is called point-convergent on setA if∀ε>0:∃Nε:∀n>Nε:∀x0∈A:|fn(x0)−f(x0)|<εf(x) is then called a uniform limit of fnfn⇉fNote: Nε only depends on ε and works for all x∈AHence uniform convergence which is much stronger than pointwise convergence

Equivalent definitions of uniform convergence #theorem

The following are equivalent:1.fn⇉f on A2.Equivalent definition via sequences3.Equivalent definition by Cauchy4.dn=supx∈A|fn(x)−f(x)|,dn→0
fn(x)=xn−x2n on [0,1]fn(x)→0supx∈[0,1]|fn(x)−f(x)|=supx∈[0,1]|xn−x2n|=supx∈[0,1](xn−x2n)=maxx∈[0,1](xn−x2n)0n−02n=01n−12n=0(xn−x2n)′=nxn−1−2nx2n−1=nxn−1(1−2xn)(xn−x2n)′=0⟺{x=0x=12n⟹dn=(12−14)=14≠0⟹fn⇉̸f

Properties of uniform limit #theorem

Let fn⇉f1.ContinuityLet ∀n:fn is continuousThen f is continuous2.IntegralLet fn be integrable on [a,b]Then f is integrable on [a,b] and ∀x∈[a,b]:∫axfn(t)dt→∫axf(t)dt3.DifferentiabilityLet fn,f be differentiableThen not necessarily fn′→f′Let ∃x0∈A:fn(x0) convergesLet ∃g:fn′⇉gThen ∃f:fn⇉f and f′=gProof for 1.Let fn⇉fLet fn be continuousLet x0∈ALet xn:xn→x0Let ε1,ε2,ε3>0:ε1+ε2+ε3=ε>0fn⇉f⟹∃N1:∀n>N1:∀x∈A:|f(xn)−fn(xn)|<ε1fn is continuous⟹∃N2:∀n>N2:∀x∈A:|fn(xn)−fn(x0)|<ε2fn→f⟹∃N3:∀n>N3:∀x∈A:|fn(x0)−f(x0)|<ε3⟹|f(xn)−f(x0)|=|f(xn)−fn(xn)+fn(xn)−fn(x0)+fn(x0)−f(x0)|≤≤|f(xn)−fn(xn)|+|fn(xn)−fn(x0)|+|fn(x0)−f(x0)|<ε⟹f is continuous