MATH 113, The three C’s: convergence, closedness and continuity Contents: 1 Real Numbers 1.1 The rationals and the decimals 1.2 The mysterious sequences of Monsieur Cauchy 1.3 Real numbers are just the decimals 2 Sequences 2.1 Convergent sequences 2.2 Cauchy-sequences and the Bolzano-Weierstrass Theorem 2.3 The Least Upper Bound Principle. Maximum vs. Supremum 2.4 Bounded and unbounded sequences 2.5 Increasing and decreasing sequences 3 Limitpoints of sequences and closed sets 3.1 Limitpoints 3.2 The Limsup and the Liminf 3.3 Closed sets 4 Continuity vs. discontinuity 4.1 Continuous functions 4.2 Uniform continuity 4.3 Discontinuities. Variation of a theme 4.4 The Big Search 4.5 Invertible functions 4.6 Continuity and closedness