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

Dedekind Theory of Irrational numbers:

  • Rational numbers, section of Rational numbers, Irrational numbers, real Numbers, Dedekind Theorem, Cantor’s Theory of Irrational numbers.
  • Cantor’s Theory, Convergent sequence of real numbers, Equivalence of the definition of Dedekind & Cantor.

Sets of Points:

  • The upper & lower bounds, l.u.b. & g.l.b. of sets, limiting point, Weierstrass Theorem, Derived sets, Countable & Non constable sets, Cardinal numbers, Open &Closed sets, Closure of a set, Perfect set, Heine-Borel Theorem.

Limit of Sequences of Real Numbers:

  • Bounded sequences, Null sequences, Monotone sequences, Convergent sequences, Fundamental theorems on limit, limitsup, limit inf of sequences, Ratio Test & other Tests, Cauchy theorems, Cauchy Convergence Criteria.

Infinite Series of Real numbers:

  • Infinite series, Tests for its convergence, Absolute convergence, Conditional convergence.

Limit of functions:

  • Concepts of Limit of functions, Limit Theorems, Some extension of Limit Concepts.

Continuity of Functions:

  • Cauchy’s  and Heine’s definitions of continuity, Properties of Continuous functions, Uniform continuity, Absolute continuity, Discontinuous Functions, Types of Discontinuities.

Differentiability:

  • Concept of Derivatives, Rolle’s theorem, Mean value theorem,  L’ Hospital Rule, Taylors.

Riemann Integration / Reimann- Stieltjes Intergral:

  • The Upper and lower R-integrals, Integrable ( R ) functions, Properties of definite and indefinite integral, Mean value theorems, Absolute convergence, convergence, Test for improper integrals. Definition & Existence of the Reimann- Stieltjes Integral & its properties.