| 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, Cauchy-Schwarz (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 Stone-Weierstrass Theorem (PDF) |
| 19 | End of Stone-Weierstrass; beginning of the theory of integration (continuous functions as uniform limits of piecewise linear functions) (PDF) |
| 20 | Riemann-Stjeltjes 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; eit= cos(t) + isin(t) (PDF) |
| 24 |
Review of series, Fourier series (PDF);
Correction (PDF)
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