Real Analysis I

A rigorous first course in analysis that rebuilds single-variable calculus on the completeness axiom: why calculus needed proof, the real numbers as a complete ordered field and the existence of the square root of 2, suprema, the Archimedean property, density and nested intervals, limits of sequences and the limit laws, monotone convergence, subsequences and the Bolzano-Weierstrass theorem, Cauchy sequences and the convergence of series with the comparison, ratio and alternating series tests, epsilon-delta limits of functions, continuity and the intermediate value theorem, compactness, the extreme value theorem and uniform continuity, the derivative and the mean value theorem, Taylor's theorem and L'Hôpital's rule, and a capstone on the Riemann integral and the fundamental theorem of calculus, with complete proofs and runnable Python examples throughout.

MATH-241 Year 2 3 credits 12 lessons · 4 h 41 min ▶ Video lessons coming soon

Offered by HSM University · Mathematics & Statistics

What you'll learn

Explain why calculus needs rigorous foundations, read and write epsilon arguments, and use the triangle inequality and its variants to estimate expressions.
State the ordered field axioms and the completeness axiom, and use completeness to prove the existence of irrational numbers such as the square root of 2.
Work with suprema and infima, prove the Archimedean property, the density of the rationals and irrationals, and the nested interval property.
Prove limits of sequences from the definition, prove and apply the algebraic and order limit theorems and the squeeze theorem, and show that limits are unique.
Apply the monotone convergence theorem, extract monotone subsequences, and prove and use the Bolzano–Weierstrass theorem.
Prove the Cauchy criterion for sequences and series, and test series for convergence with the comparison, ratio, root and alternating series tests, distinguishing absolute from conditional convergence.
Prove limits of functions with epsilon–delta arguments, use the sequential criterion to prove limit laws and to show that limits do not exist, and handle one-sided and infinite limits.
Prove that functions are continuous, classify discontinuities, and prove and apply the intermediate value theorem, including the bisection method and the continuity of inverse functions.
Characterize compact subsets of the real line, prove the extreme value theorem, distinguish continuity from uniform continuity, and prove the Heine–Cantor theorem.
Prove the rules of differentiation including the chain rule, prove Rolle's theorem and the mean value theorem, and use them to derive monotonicity, inequalities and Lipschitz bounds.
Prove Taylor's theorem with Lagrange remainder and L'Hôpital's rule, compute guaranteed error bounds, and recognize smooth functions that are not analytic.
Define the Riemann integral through Darboux sums, prove that continuous and monotone functions are integrable, and prove both parts of the fundamental theorem of calculus.

Course content

12 lessons · 4 h 41 min total · final exam

1 Why analysis: from calculus to proof 21 min · quiz
2 The real numbers: ordered fields and the completeness axiom 21 min · quiz
3 Suprema, the Archimedean property and density 21 min · quiz
4 Sequences and their limits 25 min · quiz
5 Monotone sequences, subsequences and the Bolzano–Weierstrass theorem 23 min · quiz
6 Cauchy sequences and series of real numbers 25 min · quiz
7 Limits of functions 27 min · quiz
8 Continuity and the intermediate value theorem 23 min · quiz
9 Compactness, extreme values and uniform continuity 23 min · quiz
10 The derivative and the mean value theorem 24 min · quiz
11 Taylor's theorem and L'Hôpital's rule 25 min · quiz
12 Capstone: the Riemann integral and the fundamental theorem of calculus 23 min · quiz
E Final exam 20 questions

Requirements

MATH-228 Introduction to Proofs

Recommended background. You can start any course at any time.

Description

Part of the Science stream, typically taken in year 2.