Mathematical Analysis II

Real Number system

Inductive set and the set of natural numbers N. Infinite set, countable set. Well ordering property of N. The integers and the rational numbers. The real numbers. The Field axioms, the positivity axioms, the archimedean property and the (order) completeness axiom. Supremum and infimum

Sequences.

Definition. Definition of convergence. Properties: sums, product and reciprocal. Comparison test. Squeeze Theorem. Convergence implies boundedness. Monotone convergence theorem. Cauchy sequence. Cauchy principle of convergence. Subsequences. Bolzano Weierstrass Theorem. Every real bounded sequence has a monotone subsequence. Sequence tending to + or - infinity. Subsets of the real numbers, the intervals. limit point or cluster point or accumulation point of a subset of R. Open and closed subsets. Closure. Sequentially compact subset of R. Heine Borel Theorem. Countable compactness and sequentially compactness.

Continuous Functions.

e - d definition of continuity. Sequence definition of continuity. Properties: Sum, product and quotient. Composition. Consequence of continuity. Continuous image of compact subset is compact. Extreme Value Theorem. Maximizer and minimizer. Continuous image of an interval is an interval. Intermediate Value Theorem. Monotone functions and continuity Inverse function and continuity. Uniform continuity. Any continuous function on a closed and bounded interval is uniformly continuous. Limits of a function. Properties: sum, product and quotient. Composition and limit. One sided limits. Squeeze Theorem for limits of functions.

Differentiable functions.

Definition of derivative. Properties: sum, product and quotient. Chain rule. Relative maximum and relative minimum. Local minimizer and local maximizer. Relative Extremum. Rolle's Theorem. Mean Value Theorem. Consequences of Mean Value Theorem. Monotone functions. First derivative test and second derivative test for relative extremum. Concavity. Concavity and derived function. Derivative of inverse function. Cauchy mean value Theorem. L' Hôpital's Rule and an analytic consequence. Taylor Theorem with remainder. Intermediate Value property of the derived function, Darboux theorem.

Integration.

Anti-derivative. Properties: additivity. Change of variable for anti-derivative. Riemann sums and Riemann integral. Lower and Upper Darboux sums. Lower and upper integrals. Refinement Lemma. Convergence of Lower and upper Darboux sums. Equivalent definitions of Riemann integrability. Any continuous function on a closed and bounded interval is integrable. Additivity. Properties of integrable functions. Convergence of Riemann sums. Mean Value Theorem for integrals. Darboux fundamental theorem of calculus. First fundamental theorem of Calculus, Second fundamental theorem of calculus. Products and modulus of integral functions. Integration by parts. Change of variable formula for Remain integrals. Second Mean Value Theorem for integrals.

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