The following table contains summaries for each lecture topic listed.
LEC #

TOPICS

1

Sets, ordered sets, countable sets (PDF)

2

Fields, ordered fields, least upper bounds, the real numbers (PDF)

3

The Archimedean principle; decimal expansion; intersections of closed intervals; complex numbers, CauchySchwarz (PDF)

4

Metric spaces, ball neighborhoods, open subsets (PDF)

5

Open subsets, limit points, closed subsets, dense subsets (PDF)

6

Compact subsets of metric spaces (PDF)

7

Limit points and compactness; compactness of closed bounded subsets in Euclidean space (PDF)

8

Convergent sequences in metric spaces; Cauchy sequences, completeness; Cauchy’s theorem (PDF)

9

Subsequential limits, lim sup and lim inf, series (PDF)

10

Absolute convergence, product of series (PDF)

11

Power series, convergence radius; the exponential function, sine and cosine (PDF)

12

Continuous maps between metric spaces; images of compact subsets; continuity of inverse maps (PDF)

13

Continuity of the exponential; the logarithm; Intermediate Value Theorem; uniform continuity (PDF)

14

Derivatives, the chain rule; Rolle’s theorem, Mean Value Theorem (PDF)

15

Derivative of inverse functions; higher derivatives, Taylor’s theorem (PDF)

16

Pointwise convergence, uniform convergence; Weierstrass criterion; continuity of uniform limits; application to power series (PDF)

17

Uniform convergence of derivatives (PDF)

18

Spaces of functions as metric spaces; beginning of the proof of the StoneWeierstrass Theorem (PDF)

19

End of StoneWeierstrass; beginning of the theory of integration (continuous functions as uniform limits of piecewise linear functions) (PDF)

20

RiemannStjeltjes integral: definition, basic properties (PDF)

21

Riemann integrability of products; change of variables (PDF)

22

Fundamental theorem of calculus; back to power series: continuity, differentiability (PDF)

23

Review of exponential, log, sine, cosine; e_{it}= cos(t) + isin(t) (PDF)

24

Review of series, Fourier series (PDF);
Correction (PDF)
