Detailed Examples and Explanations in Real Analysis
1. Limits and Convergence
Epsilon-Delta Definition of Limits
Example: Prove that \( \lim_{x \to 2} (3x + 1) = 7 \).
Solution: For every \( \epsilon > 0 \), we need to find \( \delta > 0 \) such that \( |x - 2| < \delta \) implies \( |(3x + 1) - 7| < \epsilon \). Simplify: \[ |(3x + 1) - 7| = |3x - 6| = 3|x - 2|. \] To ensure this is less than \( \epsilon \), choose \( \delta = \epsilon / 3 \). Thus, \( |x - 2| < \delta \) guarantees \( |(3x + 1) - 7| < \epsilon \).
Uniform vs. Pointwise Convergence
Example: Consider the sequence of functions \( f_n(x) = x^n \) on \([0, 1]\).
Pointwise Convergence: For each fixed \( x \in [0, 1) \), \( f_n(x) \to 0 \), and for \( x = 1 \), \( f_n(x) = 1 \). Thus, \( f_n(x) \to f(x) \), where: \[ f(x) = \begin{cases} 0, & x \in [0, 1) \\ 1, & x = 1 \end{cases}. \] Uniform Convergence: \( f_n(x) \) does not converge uniformly because the convergence rate depends on \( x \), and no single \( N \) works for all \( x \) within a fixed \( \epsilon \)-tolerance.
2. Metric Spaces and Topology
Open and Closed Sets
Example: In \( \mathbb{R} \) with the standard metric, prove that \( (0, 1) \) is open and \( [0, 1] \) is closed.
Solution: - \( (0, 1) \) is open because for any \( x \in (0, 1) \), we can find \( \delta > 0 \) such that \( B_\delta(x) \subset (0, 1) \). - \( [0, 1] \) is closed because its complement \( \mathbb{R} \setminus [0, 1] = (-\infty, 0) \cup (1, \infty) \) is open.
Compactness
Example: Prove that \([0, 1]\) is compact.
Solution: By the Heine-Borel theorem, \([0, 1]\) is compact because it is closed and bounded. Contrast this with \( (0, 1) \), which is bounded but not closed, and therefore not compact.
3. Differentiation and Integration
Dominated Convergence Theorem
Example: Let \( f_n(x) = \frac{\sin(nx)}{n} \) on \([0, \pi]\).
Solution: - \( f_n(x) \to 0 \) pointwise. - Dominating function: \( |f_n(x)| \leq \frac{1}{n} \), and: \[ \int_0^\pi \frac{1}{n} \, dx = \frac{\pi}{n}. \] By the Dominated Convergence Theorem: \[ \int_0^\pi f_n(x) \, dx \to 0. \]
Pathological Functions
Example: The Weierstrass function: \[ W(x) = \sum_{n=0}^\infty a^n \cos(b^n \pi x), \] where \( 0 < a < 1 \) and \( b \) is an odd integer, is continuous everywhere but differentiable nowhere.
4. Sequences and Series of Functions
Uniform Convergence
Example: The series \( \sum_{n=1}^\infty \frac{x^n}{n^2} \) converges uniformly on \([0, 1]\).
Solution: Use the Weierstrass \( M \)-test. Let \( M_n = \frac{1}{n^2} \), which satisfies: \[ \sum M_n = \sum \frac{1}{n^2} < \infty. \] Thus, the series converges uniformly.
Power Series and Radius of Convergence
Example: For the series \( \sum_{n=0}^\infty \frac{x^n}{n!} \), find the radius of convergence.
Solution: Use the ratio test: \[ \lim_{n \to \infty} \frac{a_{n+1}}{a_n} = \lim_{n \to \infty} \frac{1/(n+1)!}{1/n!} = 0. \] Thus, the radius of convergence is \( \infty \).
5. Abstract Concepts
Cantor Set
Example: Construct the Cantor set by removing the middle third of \([0, 1]\), then repeating this process for each remaining segment.
The resulting set: - Has measure zero. - Is uncountable.
Mastering proofs in real analysis can be challenging, but certain tricks and strategies can significantly enhance your understanding and ability to construct rigorous arguments. Here are some effective techniques:
---### 1. Understand Definitions Deeply - Definitions are the foundation of real analysis. Ensure you have a precise understanding of key terms like limits, continuity, compactness, and uniform convergence. - Break definitions into smaller parts to internalize their structure. - Example: The definition of continuity \(f \text{ is continuous at } c \iff \forall \epsilon > 0, \exists \delta > 0 \text{ s.t. } |x-c| < \delta \implies |f(x)-f(c)| < \epsilon\). - Know what each quantifier and inequality means intuitively.
---### 2. Work with Examples - Construct examples that satisfy or violate a given hypothesis. This can clarify how definitions and theorems behave. - Example: For uniform convergence, compare \(f_n(x) = x^n\) on \([0, 1]\) with \(f_n(x) = \frac{x}{n}\). - Ask yourself: What is an edge case? What happens at boundaries or extremes?
---### 3. Master Quantifiers - Understand how to manipulate "for all" (\(\forall\)) and "there exists" (\(\exists\)) quantifiers. - If proving \(\forall x \exists y\), start by fixing an arbitrary \(x\) and show how to find \(y\). - To disprove \(\forall x, P(x)\), find a counterexample where \(P(x)\) fails.
---### 4. Work Backwards - When proving a statement, start with the desired conclusion and think about what conditions would make it true. - Example: If proving \(A \subseteq B\), start by assuming \(x \in A\) and try to show \(x \in B\).
---### 5. Use Epsilon-Delta Arguments Carefully - Practice manipulating inequalities with \(\epsilon > 0\) and \(\delta > 0\). - Pro tip: When struggling with \(\epsilon\)-\(\delta\), write the target inequality (e.g., \(|f(x)-L| < \epsilon\)) and work backwards informally to find a candidate for \(\delta\). - Then, verify the choice works.
---### 6. Leverage Theorems - Familiarize yourself with standard theorems (Intermediate Value Theorem, Bolzano-Weierstrass, etc.) and their proofs. These can often guide your reasoning. - Example: If proving a sequence is bounded, think of the Bolzano-Weierstrass Theorem. - Use contrapositive forms of theorems when direct proofs seem challenging.
---### 7. Learn Common Proof Techniques - Direct Proof: Start from assumptions and use logical steps to reach the conclusion. - Proof by Contradiction: Assume the negation of what you want to prove and derive a contradiction. - Proof by Induction: Useful for sequences or statements involving natural numbers. - Proof by Counterexample: Disprove a universal claim by providing a single counterexample.
---### 8. Visualize Concepts - Graphical representations can often make abstract ideas clearer. - Example: Sketch sequences, functions, or sets to understand convergence, continuity, or compactness.
---### 9. Practice Constructive Thinking - For existence proofs (\(\exists\)), try to explicitly construct the required object. - Example: Prove that there exists a continuous function on \([0, 1]\) that is not differentiable by constructing \(f(x) = x^2 \sin(1/x)\).
---### 10. Break Down Complex Problems - Divide a proof into smaller claims or lemmas. Prove each part separately. - Example: To prove a sequence converges, first show it is bounded, then show it is monotonic.
---### 11. Analyze Counterexamples - Study counterexamples to understand why specific hypotheses are necessary. - Example: Continuous functions are not
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