Real Analysis Study Guide — Definitions, Proofs & Practice

Real Analysis — Comprehensive Study Guide

A compact, proof-driven roadmap: definitions, proof patterns, core theorems, worked examples, practice schedule, rubric, and resources.

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Foundations: Definitions & Core Concepts

Real Analysis is about precise language. Master the definitions first — limits, sequences, supremum/infimum, completeness, continuity, and Riemann integrability.

Action: Rewrite each definition in your own words. Practice aloud: "What does convergence mean?" "What is the completeness axiom?"

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  Limit of a sequence (ε—N formulation), limit of a function (ε—δ formulation).
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  Supremum/Infimum and why completeness of ℝ matters.
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  Continuity vs. uniform continuity; pointwise vs. uniform convergence.
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  Riemann integrability basics (upper/lower sums, integrability criteria).

Photo: study notes and textbook — use for definition recall practice.

Checkpoint: Can you explain—out loud—why the completeness of ℝ is essential for Bolzano–Weierstrass and Cauchy sequences?

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Common Proof Patterns (Templates)

Use templates to start proofs. Always structure: Given, To Prove, Strategy.

ε—δ Template (limit of function)

1. Given ε > 0, choose δ = δ(ε) (determine by reverse engineering).
2. Assume 0 < |x − a| < δ and show |f(x) − L| < ε by bounding terms (triangle inequality, etc.).
3. Conclude: for every ε there exists δ s.t. implication holds: limit proven.

Sequential Characterization

To prove continuity at a point, show: for every sequence x_n → a, f(x_n) → f(a). Conversely, use sequences to show failure of continuity.

Proof by Contrapositive / Contradiction

When direct proof is awkward, restate the negation and proceed with contrapositive for clarity.

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  Compactness proofs: open covers → finite subcover or subsequence extraction (Bolzano–Weierstrass).
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  ε/2 trick for sums and triangle inequality manipulations.
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  Typical δ choices: min(1, ε/M), or δ = ε/(|a factor| + 1) after bounding denominator.

Photo: handwritten scratchwork — emulate this 'reverse engineer' approach when discovering δ.

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Key Theorems & Series

These theorems unlock many proofs. Learn the statement, hypotheses, proof sketch, and typical applications.

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  Intermediate Value Theorem (IVT) & Mean Value Theorem (MVT) — hypotheses matter.
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  Bolzano–Weierstrass and Heine–Borel (compactness in ℝ^n).
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  Monotone Convergence Theorem, Dominated Convergence Theorem (measure-theoretic light touch).
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  Series tests: comparison, limit comparison, ratio, root; uniform convergence of series of functions.

Photo: blackboard theorem map — make your own theorem table (statement, proof idea, example).

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Worked Example — ε–δ Proof Walkthrough

Example: Prove \(\displaystyle \lim_{x\to 2} (3x+1) = 7\) using ε—δ. We show discovery (scratch work), formal proof, and common mistakes.

Scratch Work (Reverse Engineering)

Goal: Given ε > 0, choose δ such that 0 < |x−2| < δ ⇒ |(3x+1) − 7| < ε.

Compute the expression: |(3x+1) − 7| = |3x − 6| = 3|x − 2|. So to make this < ε, we need 3|x − 2| < ε ⇒ |x − 2| < ε/3. Thus choose δ = ε/3.

Formal Proof

Proof:
Let ε > 0. Choose δ = ε / 3.
Assume 0 < |x − 2| < δ. Then
  |(3x + 1) − 7| = |3x − 6| = 3|x − 2| < 3δ = 3(ε/3) = ε.
Therefore, for every ε > 0 there exists δ = ε/3 such that 0 < |x−2| < δ implies |(3x+1)−7| < ε.
Hence lim_{x→2} (3x+1) = 7. □

Common Mistakes

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  Choosing δ = ε (incorrect scaling) — forget the factor of 3.
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  Assuming x is exactly 2 somewhere in the algebra — maintain 0 < |x−2| < δ.

Photo: stepwise algebra — mimic scratch → formal flow for every ε—δ proof.

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Practice Schedule (12-week template)

A concrete weekly plan — adjust pacing to your course. Daily practice: 25–30 minute focused blocks for proofs with brief reflection.

Weeks	Focus	Daily Tasks
1–2	Sequences & Limits	5 problems/day: 3 computations, 2 short proofs (ε–N / ε–δ basics).
3–4	Continuity & Compactness	ε–δ proofs, counterexamples for pointwise vs. uniform; 1 compactness proof/week.
5–6	Series & Convergence Tests	Practice comparison, ratio, root tests; alternating series; absolute vs conditional convergence.
7–8	Differentiation & MVT	Proofs of derivative rules, MVT applications, L’Hôpital exercises.
9–10	Integration & Riemann Sums	Compute sample integrals; practice ε–δ partitions; convergence of sequences of functions.
11–12	Mixed Proofs & Review	Timed mini-exams; revisit errors; review theorem statements and proof templates.

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Rubric / Self-Assessment

Evaluate your proofs against this structured rubric. Aim for clarity, logical flow, completeness, and correctness.

Clear restatement of definitions/assumptions

Correct logical structure & strategy chosen

All algebraic/manipulative steps justified

Proper notation & statement of quantifiers (ε, δ, N, ∀, ∃)

Edge cases or counterexamples discussed if needed

Conclusion explicitly matches goal / theorem statement

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AI Tools & Online Helpers

Use these tools to accelerate learning, visualize concepts, or verify small steps — but don’t rely on them for full proofs.

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  Wolfram Alpha — check limits, derivatives, series convergence. wolframalpha.com
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  Desmos Graphing Calculator — visualize functions, sequences, continuity. desmos.com
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  ProofWiki — reference proofs, theorems, and templates. proofwiki.org
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  ChatGPT / Gemini / Claude — get hints, structured outlines, or step explanations.
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  MIT OCW Analysis Lectures — full course with lectures and exercises. ocw.mit.edu

Photo: AI-assisted study — use interactive tools to reinforce understanding and visualization.

Tip: Use AI to check logic steps or explore examples, not to replace writing proofs yourself. Practice builds intuition.