Rigorous foundations of limits, continuity, and the real number system
← Back to MathematicsBuild rigorous foundations with the construction and properties of real numbers.
Master the fundamental concept of convergence for sequences of real numbers.
Develop rigorous understanding of limits and continuous functions.
Study derivatives with complete rigor and their fundamental properties.
Master the Riemann integral and fundamental theorem of calculus.
Study convergence of function sequences and power series expansions.
Generalize analysis concepts to abstract metric spaces.
Explore topological properties of the real line and Euclidean spaces.
Extend analysis to functions of several variables with rigorous treatment.
Explore advanced concepts connecting to modern analysis and applications.
Establish the rigorous foundations of real analysis through the construction and properties of real numbers.
Understand the axiomatic approach to defining real numbers. Learn how reals can be constructed from rationals using Dedekind cuts or Cauchy sequences of rationals.
Master the field axioms for addition and multiplication: associativity, commutativity, distributivity, and existence of identities and inverses.
Study the order axioms that make ℝ an ordered field. Understand trichotomy, transitivity, and compatibility of order with field operations.
Learn the least upper bound property that distinguishes ℝ from ℚ. Understand how completeness enables the fundamental theorems of analysis.
Prove and apply the Archimedean property: for any real numbers x and y with y > 0, there exists a positive integer n such that ny > x.
Understand that both rationals and irrationals are dense in ℝ. Learn to prove density using the Archimedean property and rational approximations.
Master the concepts of least upper bound (supremum) and greatest lower bound (infimum). Learn to find and prove suprema and infima for various sets.
Prove the nested interval theorem and understand its connection to completeness. Use it to prove the Bolzano-Weierstrass theorem for bounded sequences.
Develop the fundamental theory of convergent sequences and their properties.
Master the epsilon-N definition of sequence convergence. Learn to prove convergence and divergence using the formal definition with quantifiers.
Prove and apply limit theorems: limits of sums, products, quotients. Understand the algebra of limits and squeeze theorem for sequences.
Study bounded sequences and their properties. Prove that convergent sequences are bounded and learn counterexamples for the converse.
Master the monotone convergence theorem: every bounded monotonic sequence converges. Apply this powerful tool to prove convergence without finding limits.
Understand subsequences and their convergence properties. Learn that convergent sequences have all subsequences converging to the same limit.
Prove that every bounded sequence has a convergent subsequence. Understand this as a fundamental compactness property of bounded sets in ℝ.
Study Cauchy sequences and prove they are equivalent to convergent sequences in ℝ. Understand how this characterization reveals completeness.
Master limit superior and limit inferior for bounded sequences. Learn their relationship to convergence and applications to series convergence tests.
Develop rigorous understanding of function limits and continuous functions.
Master the epsilon-delta definition of function limits. Learn to construct proofs of limits and understand the logical structure of the definition.
Extend limit concepts to one-sided limits and limits involving infinity. Understand their relationship to standard limits and applications.
Prove limit theorems for sums, products, quotients, and compositions. Learn the sequential characterization of function limits.
Understand continuity at a point and on intervals. Learn equivalent characterizations: epsilon-delta, sequential, and via limits.
Classify discontinuities as removable, jump, or essential. Understand the structure of discontinuity sets for monotonic functions.
Prove the Intermediate Value Theorem and understand its topological significance. Apply it to root-finding and proving existence of solutions.
Prove that continuous functions on closed bounded intervals attain their maximum and minimum. Understand the role of compactness.
Distinguish uniform continuity from pointwise continuity. Prove that continuous functions on compact sets are uniformly continuous.
Study derivatives with complete rigor and prove the fundamental theorems of differential calculus.
Master the limit definition of derivative. Understand differentiability as a local linear approximation property and geometric interpretation.
Prove that differentiable functions are continuous. Understand examples of continuous but non-differentiable functions like |x|.
Prove Rolle's theorem and understand its geometric meaning. Learn its role as a stepping stone to the Mean Value Theorem.
Prove the Mean Value Theorem and understand its fundamental importance. Apply it to prove monotonicity criteria and other results.
Prove L'Hôpital's rule for indeterminate forms. Understand the conditions for its application and common pitfalls in usage.
Prove Taylor's theorem with various forms of remainder. Understand the role of higher-order derivatives in approximation theory.
Prove the inverse function theorem for real functions. Understand the conditions for existence and differentiability of inverse functions.
Apply differentiation to optimization problems. Understand first and second derivative tests and their limitations.
Master the Riemann integral and prove the fundamental theorem of calculus.