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• Number Systems: Ordered fields
• rational, real and complex numbers
• Archimedean property
• supremum, infimum and completeness
• Convergence, completeness, completion of real numbers
• open sets, closed sets, compact sets,
• Heine Borel theorem, connected sets.
• Sequences and Series of Real Numbers
• Limits of sequences, algebra of limits.
• Bolzano Weierstrass theorem, Cauchy sequences, liminf, limsup
• Limits of series, convergences tests, absolute and conditional convergence, power series
• Continuity: Functions, continuity and compactness, existence of minimizers and maximizers
• uniform continuity, continuity and connectedness, intermediate mean value theorem
• Monotone functions and discontinuities.
• Differentiation: Mean value theorem, L’Hopital’s Rule, Taylor’s theorem