Numbers, Limits and the Integral

Contents

1. 1
   The Natural Numbers

   Five axioms, what each one rules out, and how induction and recursion build all of arithmetic.

   54 min read

2. 2
   Sets, Functions and Equivalence

   The language of sets and maps, and quotients: treating different things as the same.

   32 min read

3. 3
   Integers and Rationals

   Building ℤ and ℚ as quotients of pairs, checking operations are well defined, and finding the gaps in ℚ.

   27 min read

4. 4
   The Real Numbers

   Completing ℚ with Cauchy sequences: the least upper bound property and the idea of completion.

   37 min read

5. 5
   Quantifiers and the Shape of a Proof

   Reading and negating ∀ε ∃δ statements, and the proof patterns used from here to Perelman.

   27 min read

6. 6
   Sequences

   Limits, lim sup and lim inf, monotone convergence and Bolzano–Weierstrass.

   27 min read

7. 7
   Series

   Convergence tests, rearrangements and the geometric series in its two lives.

   25 min read

8. 8
   Infinite Sets

   Countable and uncountable sets, the diagonal argument, and where the axiom of choice will be needed.

   18 min read

9. 9
   Continuous Functions

   Continuity, the intermediate value theorem, uniform continuity, and the first maximum principle.

   27 min read

10. 10
    Derivatives

    The mean value theorem, the second-derivative test, and the first finite-time blow-up.

    30 min read

11. 11
    The Riemann Integral

    Upper and lower sums, the fundamental theorem, integration by parts, and where Riemann’s integral fails.

    30 min read

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