Alphabeta MathSequences and Series of Functions; Uniform Convergence: Examples and Counterexamples
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1 · Prerequisites
Abstract Algebra
Linear Algebra
Real Analysis
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Properties of the Integral and the Working FTC
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Suprema and Infima
- The Derivative and the Mean Value Theorems
- The Riemann Integral: Definition and Integrability
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Counterexample
fₖ(x)=xᵏ⁺¹ converges pointwise but not uniformly on [0,1]
Counterexample
Shrinking rectangles converge pointwise to zero while every integral equals one
Counterexample
Dini's theorem fails on [0,∞): x/(ι(k+1)+x) decreases pointwise to zero but not uniformly
Counterexample
Dini's theorem fails for discontinuous approximants: shrinking interval indicators decrease pointwise to zero but not uniformly
Counterexample
Dini's theorem fails for a discontinuous limit: powers on [0,1] decrease pointwise to a discontinuous endpoint indicator but not uniformly
Counterexample
Continuous triangular spikes on [0,1] converge pointwise to zero but not uniformly when monotonicity is absent
Counterexample
The double sequence (m+1)/(m+n+2) has unequal iterated limits
Counterexample
x/(1+(k+1)²x²) converges uniformly to zero on ℝ while every derivative at zero equals one
Sources
- J. Lebl, Basic Analysis I, §6.1
- W. Trench, Introduction to Real Analysis
- William Faris, Real Analysis: Part I, §13.2
- Dini's theorem (Wikipedia)
- Stephen Abbott, Understanding Analysis, 2nd ed., Exercise 2.3.13
- Trinity College Dublin MA2223, Homework 3 Solutions
- King Saud University, Final Exam Model Solution