Real Analysis

Units

1. 1

   The real numbers

   3 lessons

   1. Ordered fields and absolute valueUse the field and order axioms of ℝ to prove facts about inequalities, prove the triangle inequality |a + b| ≤ |a| + |b|, and read |x − a| < ε as a band on the number line.
   2. Suprema and completenessDefine upper bounds and the supremum, find sup and inf of sets such as {1 − 1/n}, state the completeness axiom, and show why ℚ is not complete: {x : x² < 2} has no rational supremum.
   3. The Archimedean property and densityProve from completeness that ℕ is unbounded in ℝ, so that for every ε > 0 some 1/n < ε, and then prove that between any two reals lie a rational and an irrational.
2. 2

   Sequences

   5 lessons

   1. Convergence of sequencesState the ε-N definition of aₙ → L, prove convergence from it by choosing N for a given ε, and prove that limits are unique, convergent sequences are bounded, and (−1)ⁿ diverges.
   2. Limit theorems for sequencesProve and use the algebraic limit theorem (sums, products, quotients), the order limit theorem and the squeeze theorem, each from the ε-N definition.
   3. Monotone sequencesProve that a bounded monotone sequence converges to its supremum (or infimum), and use it on the recursion aₙ₊₁ = √(2 + aₙ).
   4. Subsequences and Bolzano-WeierstrassUse subsequences to detect divergence, and prove by halving intervals that every bounded sequence has a convergent subsequence.
   5. Cauchy sequencesDefine a Cauchy sequence, prove that in ℝ Cauchy and convergent mean the same, and see why consecutive terms getting close is not enough.
3. 3

   Series

   2 lessons

   1. Convergence of seriesA series converges when its partial sums do; the Cauchy criterion, the divergence test, the comparison test and the harmonic series by grouping, with the partial sums graphed.
   2. Absolute convergence and rearrangementsAbsolute convergence implies convergence; the alternating harmonic series converges conditionally, and Riemann's theorem lets a rearrangement reach any sum.
4. 4

   Topology of the real line

   2 lessons

   1. Open and closed setsNeighbourhoods, open sets, limit points, closed sets as those containing their limit points, and the closure, with intervals and {1/n} as examples.
   2. Compact setsA set is compact when every sequence in it has a subsequence converging in it; compact sets of R are the closed bounded ones (Heine-Borel), and open covers.
5. 5

   Limits and continuity

   4 lessons

   1. Limits of functionsThe ε-δ limit and the sequential criterion, used to show that sin(1/x) has no limit at 0, with the graph oscillating inside every band.
   2. Continuous functionsContinuity by ε-δ and by sequences, the algebra of continuous functions and composition, and Dirichlet's function continuous nowhere.
   3. Continuous functions on compact setsA continuous function on [a, b] attains a maximum and a minimum (extreme value theorem) and every value between f(a) and f(b) (intermediate value theorem), with bisection drawn.
   4. Uniform continuityOne δ for every point: 1/x on (0, 1) is continuous but not uniformly, and every continuous function on a compact set is uniformly continuous.
6. 6

   Differentiation

   3 lessons

   1. The derivativeThe derivative as a limit of difference quotients, differentiable implies continuous but |x| shows the converse fails, and the product and chain rules proved.
   2. The mean value theoremRolle's theorem from the extreme value theorem, the mean value theorem by tilting, and its consequences: f' = 0 means constant, f' > 0 means increasing.
   3. Taylor's theoremTaylor polynomials with the Lagrange remainder, bounding the error of approximating eˣ and sin x on an interval, the curves and the error band graphed.
7. 7

   The Riemann integral

   3 lessons

   1. Upper and lower sumsPartitions, upper and lower Darboux sums drawn as rectangles above and below x², refinements, and the integral as the common value when sup L = inf U.
   2. Integrable functionsThe criterion U − L < ε, continuous and monotone functions are integrable, Dirichlet's function is not, and linearity and order of the integral.
   3. The fundamental theorem of calculusBoth parts proved: ∫ₐᵇ f' = f(b) − f(a) from the mean value theorem, and F(x) = ∫ₐˣ f is continuous and differentiable where f is continuous.
8. 8

   Sequences and series of functions

   3 lessons

   1. Pointwise and uniform convergencexⁿ on [0, 1] converges pointwise to a discontinuous limit; uniform convergence asks one N for every x, a tube of width ε around the limit.
   2. Consequences of uniform convergenceA uniform limit of continuous functions is continuous, integrals pass to the limit, and the Weierstrass M-test proves uniform convergence of a series.
   3. Power seriesThe radius of convergence, uniform convergence on closed subintervals, and term-by-term differentiation and integration, worked on Σ xⁿ = 1/(1 − x).