Real Analysis

MATH370 (Analysis I)

1. 0 Extra Notes
• [Rudin] 1.11: Suppose \(S\) is an ordered set with the least-upper-bound-property, \(B \subset S\), \(B\) is not empty, and \(B\) is bounded below. Let \(L\) be the set of all lower bounds of \(B\). Then \(\alpha = \sup L\) exists in \(S\), and \(\alpha = \inf B\).
• [Rudin] 1.20(a): [Archimedean Property] \(\mathbb{N}\) is unbounded.
2. 1 Real Number System
• [Lecture 1] \(\mathbb{R}\) is complete
• [Lecture 1] \(\sqrt{2}\) is irrational
3. 2 Completeness Axiom
• [Lecture 2] Suppose that \(E \subset \mathbb{Z}\) is non-empty and bounded. Then, \(\sup E \in E\)
• [Lecture 2] [Archimedean Principle] \(\mathbb{N}\) is unbounded. For all \(M > 0\), there exists \(n \in \mathbb{N}\) such that \(n > M\).
• [Lecture 2] \(\sqrt{2}\) exists. It is \(s = \sup \{x \in \mathbb{R}^{+} \mid x^2 < 2\}\) where \(s^2 = 2\).
• [Lecture 2] \(\mathbb{Q}\) is dense in \(\mathbb{R}\)
• [Lecture 2] \(\mathbb{R}-\mathbb{Q}\) is dense in \(\mathbb{R}\) [TODO]
4. 3 Well-Ordering Principle
• [Lecture 3] The Well-Ordering Principle
• [Lecture 3] Principle of Mathematical Induction [TODO]
• [Lecture 3] \(\sqrt{n}\) is irrational [TODO]
5. 4 Countable and Uncountable Sets
6. 5 Limit of Sequences
• [WW] 2.1.1b: Prove that \(1 + \pi/\sqrt{n} \rightarrow 1\) as \(n \rightarrow \infty\).
• [Lecture 5] Prove that \(\lim\limits_{n \rightarrow \infty} \frac{2n^2 + 1}{n^2 - n - 10} = 2\).
• [Lecture 5] Every convergent sequence is bounded.
• [Lecture 5] \(x_n\) converges to \(x\) if and only if every subsequence of \(x_n\) converges to \(x\).
• [Lecture 5] Show that \(((-1)^n)_n = (-1,1,-1,1,...)\) diverges.
7. 6 Limit Theorems
• [Lecture 6] \(\lim (x_n + y_n) = x + y\).
• [Lecture 6] \(\lim (\alpha x_n) = \alpha x\) for all \(\alpha \in \mathbb{R}\).
• [Lecture 6] \(\lim (x_nx_n) = xy\).
• [Lecture 6] \(\lim (x_n/y_n) = x/y\) if \(y \neq 0\).
• [Lecture 6] [Squeeze Theorem] If \(x_n \leq y_n \leq z_n\) for all \(n \in \mathbf{N}\), and if \(\lim x_n = \lim z_n = l\), then \(\lim y_n = l\).
• [Lecture 6] Suppose \(E\) has a finite supremum, then there exists a sequence \(x_n\) such that \(\lim\limits_{n \to \infty} x_n = \sup(E)\) [TODO]
• [Lecture 6] [Comparison Theorem] Let \(x_n\) and \(y_n\) be two convergent sequences with limits \(x\) and \(y\) respectively. If there exists \(N_0\) such that \(x_n \leq y_n\) for all \(n \geq N_0\), then \(x \leq y\).
• [Lecture 6] Divergence Definition and Theorem.
8. 7 The Monotone Convergence Theorem
• [Lecture 7] If a sequence is monotone and bounded, then it converges.
• [Lecture 7] Monotone Convergence Theorem - Example 1, \(a_n = a^n\).
• [Lecture 7] Monotone Convergence Theorem - Example 2, \(x_{n+1} = \sqrt{2 + x_n}\).
• [Lecture 7] Monotone Convergence Theorem - Example 3, \(a_n = a^{1/n}\).
• [Lecture 7] Monotone Convergence Theorem - Example 4, \(x_{n+1} = \frac{x_n + \frac{2}{x_n}}{2}\). Find \(\lim\limits_{n \rightarrow \infty} x_n\)
• [Lecture 7] [The Nested Interval Property]
• [Lecture 7] [Bolzano-Weierstrass] Every bounded sequence contains a convergent subsequence.
9. 8 Cauchy Sequences
• [Lecture 8] Every convergent sequence is a Cauchy sequence
• [Lecture 8] Cauchy Sequences are Bounded
• [Lecture 8] A sequence converges if and only if it is a Cauchy sequence.
• [Lecture 8] Example 1
10. 9 Limits of Functions
• [Lecture 9] Basic Definitions and Theorems
• [Lecture 9] Local Boundedness Lemma
• [Lecture 9] Sequential Criterion for Functional Limits
• [Lecture 9] Algebraic Limit Theorem for Functional Limits
• Example 1: Prove that \(\lim\limits_{x \rightarrow 2} 3x + 1 = 7\)
• Example 2: Prove that \(\lim\limits_{x \rightarrow 2} x^2 = 4\)
• [Lecture 9] Example 3: Show that \(\lim\limits_{x \rightarrow 0} \frac{1}{\sin x}\) doesn't exist.
11. 10 Continuity
• [Lecture 10] Definitions
• [Lecture 10] Show that \(f(x)=1/x\) is not continuous at \(x = 0\).
• [Lecture 10] Is \(x\sin\left(\frac{1}{x}\right)\) is continuous at \(x=0\)?
• [Lecture 10] Dirichlet's Function
• [Lecture 10] Sequential Characterization of Continuity
• [Lecture 10] Extreme Value Theorem
• [Lecture 10] Intermediate Value Theorem
• [Lecture 10] IVT Corollary: The image of a bounded and closed interval of a continuous function is an interval
• [Lecture 10] IVT: Fixed Point Theorem
• [Lecture 10] The Thomae/Popcorn Function
12. 11 Uniform Continuity
• [Lecture 11] Definitions
• [Lecture 11] Prove that \(f(x) = x^2\) is uniformly continuous on \([0,M]\) where \(M < \infty\) but not uniformly continuous on \(\mathbb{R}\)
• [Lecture 11] \(\frac{1}{x}\) is continuous on \((0,1)\) but not uniformly continuous on \((0,1)\).
• [Lecture 11] [Uniform Continuity Theorem] If \(f:I \rightarrow \mathbb{R}\) is continuous on a closed and bounded interval, then \(f\) is uniformly continuous.
• [Lecture 11] Suppose \(f:E \rightarrow \mathbb{R}\) is uniformly continuous. If \(x_n \in E\) is Cauchy, then \(f(x_n)\) is Cauchy.
• [Lecture 11] [Continous Extension]
13. 12 Differentiability
• [Lecture 12] Definitions
• [Lecture 12] [Differentiability Theorems] (Product Rule, Chain Rule, ...)
• [Lecture 12] If \(f\) is differentiable at a point \(x = a\), then \(f\) must be continuous at \(x = a\).
• [Lecture 12] Example: Show that \(f(x)=|x|\) is not differentiable at \(x=0\).
• [Lecture 12] Example: Show that \(f\) is not differentiable at \(x=0\).
14. 13 Mean Value Theorem
• [Lecture 13] Rolle's Theorem
• [Lecture 13] Mean Value Theorem
• [Lecture 13] Definition of increasing and decreasing functions
• [Lecture 13] [Theorem 0.4] If \(f'(x) > 0\) for all \(x \in (a,b)\), then \(f\) is strictly increasing on \((a,b)\).
• [Lecture 13] [Theorem 0.4] If \(f\) is increasing on \((a,b)\), then \(f'(x) \ge 0\) for all \(x \in (a,b)\).
• [Lecture 13] [Theorem 0.4] If \(f'(x) = 0\) for all \(x \in (a,b)\), then \(f\) is a constant function.
• [Lecture 13] \(e^x \geq 1 + x\) for all \(x \geq 0\).
• [Lecture 13] \(\sin(x) \leq x\) for all \(x \geq 0\).
15. 14 Riemann Integration
• [Lecture 14] Definitions (Upper/Lower Riemann Sums, Darboux Integrable)
• [Lecture 14] If \(f\) is Darboux integrable, then \((U) \int_a^b f(x)dx = (L) \int_a^b f(x)dx\)
• [Lecture 14] The Dirichlet function \(f\) is not Darboux integrable on any interval \([a,b]\).
• [Lecture 14] A discontinuous function can be Darboux integrable
• [Lecture 14] Definitions (Riemann Integrable)
• [Lecture 14] [Lemma 0.9] If a function \(f\) is Reimann integrable on \([a,b]\), then \(f\) is bounded on \([a,b]\).
• [Lecture 14] [Theorem 0.10] A function is Darboux integrable if and only if it is Reimann integrable. Moreover, \(I(f) = \int_a^b f(x)dx\).
16. 15 Properties of Riemann Integration
• [Lecture 15] Properties of the Riemann Integration: Linearity and Disjoint Additivity
• [Lecture 15] Suppose that \(f(x) \leq g(x)\) for all \(x \in [a,b]\). Then \(\int_a^b f(x) \ dx \leq \int_a^b g(x) \ dx\)
• [Lecture 15] Suppose that \(f\) be Riemann integrable on \([a,b]\). Then, \(|f|\) is Riemann integrable.
17. 16 Fundamental Theorem of Calculus
• [Lecture 16] Fundamental Theorem of Calculus

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Self Study Older Notes (To be Reviewed)