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The exhaustive list of topics in Modern Analysis in which we provide Help with Homework Assignment and Help with Project is as follows:

• Sets: Sets of numbers, set operations, family of subsets, constructing
• new sets from old ones, De Morgan’s laws.
• Maps: Surjection, injection, invertible maps, direct and inverse images
• and their relations, inverse of a map, composition of maps, restriction
• of a map.
• Finite and infinite sets, countable and uncountable sets, Results related
• to countable sets.
• Countability of rational numbers.
• Absolute value of a real number, epsilon neibourhood of a point,
• supremum and infimum and properties. Completeness property of the
• real numbers. Archimedean property, Density of rational numbers.
• Nested intervals property, uncountability of reals..
• Sequences, convergence, examples.
• Conditions of convergence, monotone sequence, examples.
• Tests of convergence of sequences of real numbers.
• Subsequences, Bolzano-Weierstrass theorem, Cauchy’s criterion.
• Infinite series, tests of convergence.
• Definition and examples of metric spaces.
• Open ball, neighborhood of a point, open set, closed set, and their
• properties.
• Interior, exterior and their properties.
• Closure of a set and their properties.
• Limit point of a subset and limit of a sequence in a metric space,
• Boundary of a set and properties.
• Equivalence of metric spaces. Product spaces.
• Continuity of a map between two metric spaces and examples.
• Various characterizations continuous maps.
• Cauchy sequence, Complete metric spaces. Example of complete and
• incomplete metric spaces.
• Completeness of the space of real numbers.
• Compactness, examples. Continuous image of a compact space.
• Heine-Borel theorem.
• Separation. Connectedness, connected subsets of the real metric space.
• Properties of connected spaces. Examples. characterization of
• connectedness. Intermediate value theorem.
• Differentiable functions, higher derivatives, Leibnitz Rule.
• Rolle’s theorem, Mean value theorems.
• L’ Hospital rule. Examples.
• Taylor’s formula and Taylor series.
• Monotone functions, maxima and minima.
• Riemann Integral, example of functions which are not Riemann
• integrable.
• Criterion of Riemann integrability, Properties of Riemann integral.
• Fundamental theorems of integral calculus.
• Integration by parts.
• Trapezoidal rule and Simpsion’s rules of approximation of a Riemann
• integrals.
• Improper integrals.
• Sequence and Series of functions, uniform convergence of functions.
• Weierstrass M-test.
• Power Series, radius of convergence.
• Properties of functions represented by real power series.
• Fourier Series.
• More on Fourier Series.