Real Analysis (1) - NU

. The Real Number System (\(\mathbb{R}\))Algebraic and Order Properties: Verifying fields, inequalities, and absolute value behaviors.The Completeness Axiom: Understanding the supremum (least upper bound) and infimum (greatest lower bound) properties.Applications of Completeness: Proving the Archimedean Property and the Density of Rational Numbers in \(\mathbb{R}\).2. Topology of the Real LineOpen and Closed Sets: Defining neighborhoods, interior points, boundary points, and limit points.Compactness: Introduction to the Heine-Borel Theorem (a subset of \(\mathbb{R}\) is compact if and only if it is closed and bounded).3. Sequences of Real NumbersLimits of Sequences: Writing formal ε-N proofs for convergent sequences.Limit Theorems: Proving squeeze theorems, Monotone Convergence Theorem, and subsequence properties.The Bolzano-Weierstrass Theorem: Proving that every bounded sequence contains a convergent subsequence.Cauchy Sequences: Utilizing the Cauchy Criterion to test for convergence without knowing the explicit limit.4. Limits and Continuity of FunctionsLimits of Functions: Formulating the rigorous ε-δ definition of a limit at a point.Continuous Functions: Proving continuity, checking for uniform continuity, and applying the Intermediate Value Theorem and Extreme Value Theorem