Strengthen your understanding of real analysis concepts through this comprehensive 4-hour exercise collection that serves as practical companion material to core real analysis theory. Work through detailed proofs and problem-solving techniques covering fundamental topics including the triangle inequality, supremums and infimums of sets, the completeness axiom, sequence convergence and Cauchy sequences, open and closed sets, compactness, and continuity proofs using epsilon-delta definitions. Practice proving continuity for various functions including polynomials, absolute value, square root, trigonometric, and exponential functions, while exploring the distinction between continuity and uniform continuity. Develop proficiency in applying the Mean Value Theorem, working with limits involving logarithmic functions, and understanding the relationships between bounded sets and their supremums and infimums. Each exercise builds upon theoretical foundations to provide hands-on experience with the rigorous proof techniques essential for mastering real analysis, making this an invaluable resource for students seeking to deepen their problem-solving skills beyond standard textbook material.

Syllabus

Proof: The General Triangle Inequality | Real Analysis
Supremum of the Union of Sets | Real Analysis
Bound for Supremum of the Intersection of Sets | Real Analysis Exercises
How Completeness Guarantees Infimums | Real Analysis
Prove Infimums Exist with the Completeness Axiom | Real Analysis
Supremum of the Union of Sets | Real Analysis
Supremums and Addition | Real Analysis Exercises
Proof: Infimum of {1/n} = 0 | Real Analysis
Proof: Supremum of {1/n} = 1 | Real Analysis
Proof: Supremum of {n/(n+1)} = 1 | Real Analysis
Proof: Sequence 1/n Converges to 0 | Real Analysis Exercises
Proof: Sequence {sqrt(n+1)-sqrt(n)} Converges to 0 | Real Analysis Exercises
Proof for Absolute Value of a Convergent Sequence | Real Analysis Exercises
Bounded Sequence and Zero Convergent Sequence Limit Law | Real Analysis Exercises
Proof for Square Root of Convergent Sequence | Real Analysis Exercises
Prove Sequence Limits with Greatest Integer Function | Real Analysis Exercises
Sequence Limits Don't Preserve Strict Inequalities | Real Analysis Exercises
Proof: Sequence (1/n) is a Cauchy Sequence | Real Analysis Exercises
Do These Cauchy Sequences Exist? | Real Analysis
Identifying Open, Closed, and Compact Sets | Real Analysis Exercises
Compact Sets Contain Supremum and Infimum | Real Analysis Exercises
Proof: f(x) = x is Continuous using Epsilon Delta Definition | Real Analysis Exercises
Proof: ax+b is Continuous using Epsilon Delta Definition
Proof: f(x) = |x| is Continuous using Epsilon Delta Definition | Real Analysis Exercises
Proof: sqrt(x) is Continuous using Epsilon Delta Definition | Real Analysis Exercises
Proof x^2 is Continuous using Epsilon Delta Definition | Real Analysis Exercises
Proof x^3 is Continuous using Epsilon Delta Definition | Real Analysis Exercises
Proof f(x)=sin(x) is Continuous using Epsilon Delta Definition | Real Analysis Exercises
Proof: e^x is Continuous using Epsilon Delta Definition | Real Analysis Exercises
Proof f(x)=x is Uniformly Continuous using Epsilon Delta Definition | Real Analysis Exercises
x^3 is Continuous but not Uniformly Continuous (Sequential Criterion) | Real Analysis Exercises
Proof sqrt(x) is Uniformly Continuous using Epsilon Delta Definition | Real Analysis Exercises
Combinations of Uniformly Continuous Functions | Real Analysis Exercises
Limit of lnx as x approaches Infinity (with Mean Value Theorem) | Real Analysis Exercises
Limit of lnx as x approaches 0 | Real Analysis Exercises
Proof: If a is less than b, then sup(A) is less than inf(B) | Real Analysis Exercises