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)
