Real Analysis
MATH370 (Analysis I)
- 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.
- 1-2 Completeness
- [Lecture 1] \(\mathbb{R}\) is complete.
- [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] \(\mathbb{Q}\) is dense in \(\mathbb{R}\).
- [Lecture 2] \(\sqrt{2}\) exists. It is \(s = \sup \{x \in \mathbb{R}^{+} \mid x^2 < 2\}\) where \(s^2 = 2\).
- [Lecture 2] \(\mathbb{R}-\mathbb{Q}\) is dense in \(\mathbb{R}\). [TODO]
- 3 Well-Ordering Principle
- 4 Countable and Uncountable Sets
- 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.
- 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] [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.
- 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. [TODO]
- 8 Cauchy Sequences
- [Lecture 8] Cauchy Sequences are Bounded [TODO]
- [Lecture 8] Every convergent sequence is a Cauchy sequence [TODO]
- [Lecture 8] A sequence converges if and only if it is a Cauchy sequence. [TODO]
Self-Study Notes (Understanding Analysis) [Old Notes, to be removed]
- 1.2 Preliminaries
- Prove that if \(p^2\) is even, then \(p\) is even.
- Prove that \(\sqrt{2}\) is irrational.
- Prove that \(\sqrt{3}\) is irrational.
- Two real numbers \(a\) and \(b\) are equal if and only if for every real number \(\epsilon > 0\) it follows that \(|a - b| < \epsilon\).
- Definition and Properties
- Let \(x \in \mathbf{R}\), prove that \(|x| \geq 0\).
- Let \(x \in \mathbf{R}\), prove that \(-|x| \leq x \leq |x|\).
- For two real numbers \(x\) and \(y\), prove that \(\max(x,y) = \frac{1}{2}(x+y+|x - y|)\).
- \(|ab| = |a|\cdot|b|\) holds for all real numbers \(a\) and \(b\).
- \(|a+b| \leq |a|+|b|\) holds for all real numbers \(a\) and \(b\).
- \(|a - c| \leq |a - b|+|b - c|\) holds for all real numbers \(a\), \(b\) and \(c\).
Exercises
- 1.3 Axiom of Completeness (Upper and Lower Bounds)
- Bounds Definitions
- [Lemma 1.3.8] If \(s\) is an upper bound for \(A\), then \(s = \sup A\) if and only if for every choice of \(\epsilon > 0\), there exists an element \(a \in A\) satisfying \(s - \epsilon < a\).
- Compute the suprema and infima of the following sets ...
- Prove that if \(a\) is an upper bound for \(A\), and if \(a\) is also an element of \(A\), then it must be that \(a=\sup A\)
- Assume that \(A\) and \(B\) are nonempty, bounded above, and satisfy \(B \subseteq A\). Show \(\sup B \leq \sup A\).
- The set defined as \(c+A = \{c + a : a \in A\}\) where \(c\) is a constant has a least upper bound equal to \(c + sup A\).
- [1.3.5] The set defined as \(cA = \{ca : a \in A\}\) where \(c \geq 0\) has \(\sup cA = c\sup A\).
- [1.3.6] The set defined as \(A + B = \{a + b: a \in A \text{ and } b \in B\}\) has \(\sup (A+B) = \sup A + \sup B\).
- [1.3.6] The set defined as \(A + B = \{a + b: a \in A \text{ and } b \in B\}\) has \(\sup (A+B) = \sup A + \sup B\) (Alternative Proof).
- [1.3.4] Let \(A\) and \(B\) be nonempty and bounded above. Find a formula for \(\sup (A \cup B)\) and prove that it is correct.
- [1.3.11] If \(A\) and \(B\) are nonempty and bounded sets of real numbers such that \(A \subseteq B\) then \(\inf B \leq \inf A \leq \sup A \leq \sup B\).
- Let \(A = \{x \in \mathbf{R}: x < 0\}\). Then \(\sup(A) = 0\).
- If \(\sup A < \sup B\), then show that there exists an element \(b \in B\) that is an upper bound for \(A\).
- Determine if the following statements are true or false.
- Let \(A\) and \(B\) be nonempty subsets of \(\mathbb{R}\) with \(A\) bounded above and \(B\) bounded below. Define \(A - B = \{a - b: a \in A, b \in B\}\). Prove that \(\sup(A - B) = \sup A - \inf B\).
- Prove that for any nonempty set \(S\) of real numbers that is bounded below, \(\inf S = -\sup(-S)\) where \(-S = \{-s: s \in S\}\).
- Let \(A\) be a nonempty subset of \(\mathbb{R}\) that is bounded above. Define \(S = \{ x \) is an upper bound for \(A\}\). Prove that \(\inf S = \sup A\).
Exercises
- 1.4 Consequences of Completeness
- 1.5 Cardinality
- 2.2 The Limit of a Sequence
- Sequences Definitions
- [Uniqueness of Limits (2.2.7)] The limit of a sequence, when it exists must be unique.
- \(\lim\big(\frac{1}{\sqrt{n}}\big)= 0\). (Show the Limit Template)
- Prove that \(\lim\big(\frac{n+1}{n}\big)= 1\).
- Prove that \(\lim\big(\frac{1}{3n^3}\big)= 0\).
- Prove that \(\lim\big(n\big)\) diverges.
- Let \(x_n \geq 0\) for all \(n \in \mathbf{N}\). Show that if \((x_n) \longrightarrow x\), Then \((\sqrt{x_n}) \longrightarrow \sqrt{x}\).
- Let \(\{a_n\}\) be a sequence. Prove that if \(\lim\limits_{n \rightarrow \infty} a_n = L\) and \(\lim\limits_{n \rightarrow \infty} a_{n+1} = M\), then \(L = M\).
- If \(\{a_n\}\) is a convergent sequence with \(\lim\limits_{n \rightarrow \infty} a_n = L\). Prove that \(\{ |a_n|\}\) converges as well. Does \(\{ |a_n|\}\) converges to \(L\)?
- Verify using the definition of convergence that the following sequences converge to the proposed limit.
- Consider the sequence \(\{a_n\}\) defined by \(a_n = \frac{(-1)^nn}{n+1}\). Determine whether this sequence converges or diverges. If it converges, find its limit.
- Let \(\{a_n\}\) be a sequence. Prove that if \(\lim\limits_{n \rightarrow \infty} |a_n| = 0\), then \(\lim\limits_{n \rightarrow \infty} a_n = 0\).
Exercises
- 2.3 The Algebraic and Order Limit Theorems
- Show that if \((|x_n|) \rightarrow 0\) for all \(n \in \mathbf{N}\), then \(x_n \rightarrow 0\).
- Argue that the sequence \(1,0,1,0,0,1,0,0,0,1,0,0,0,0,1,...\) does not converge to zero. For what values of \(\epsilon > 0\) does there exist a response \(N\)? For which values of \(\epsilon > 0\) is there no suitable response.
- Informally speaking, the sequence \(\sqrt{n}\) "converges to infinity". (a) Create a rigorous definition for the statement \(\lim\limits_{n \rightarrow \infty} x_n = \infty\) ...
- Create a rigorous definition for the statement \(\lim\limits_{n \rightarrow \infty} x_n = -\infty\). Give an example of a sequence that satisfies this definition.
- Construct a sequence \(\{a_n\}\) such that \(a_n > 0\) for all \(n \in \mathbb{N}\), \(\lim\limits_{n \rightarrow \infty} a_n = 0\), but \(\lim\limits_{n \rightarrow \infty} \frac{1}{a_n} = \infty\).
- Prove or disprove: If \(\lim\limits_{n \rightarrow \infty} a_n = 0\) and \(\{b_n\}\) is a bounded sequence, then \(\lim\limits_{n \rightarrow \infty} a_nb_n = 0\).
Exercises
- 2.4 The Monotone Convergence Theorem / 2.5 Bolzano Weierstrass
- 2.7 Properties of the Infinite Series
- Series Definitions
- (2.4.6) Cauchy Condensation Test
- (2.7.1) The Algebraic Limit Theorem for Series
- (2.7.2) Cauchy Criterion for Series
- (2.7.3) If the series \(\sum_{n=1}^{\infty} a_n\) converges then \((a_n) \rightarrow 0\).
- (2.7.4) Comparison Test for Series
- (2.7.5) The Geometric Series
- (2.7.6) [The Absolute Convergence Test] If the series \(\sum_{n=1}^{\infty}|a_n|\) converges, then \(\sum_{n=1}^{\infty} a_n\) converges as well.
- (2.7.7) The Alternating Series Test
- 3.1 Topology of R
- 3.2 Open and Closed Sets
- Open Sets
- Limit Points
- Closed Sets
- [3.2.3] The union of an arbitrary collection of open sets is open and the intersection of a finite collection of open sets is open.
- [Closed Sets] Prove that a set \(F \subseteq \mathbf{R}\) is closed if and only if it contains its limit points.
- [Closed Sets] Prove that a closed interval \([c,d] = \{c \leq x \leq d\}\) is a closed set.
- Closure
- [3.2.13] A set \(O\) is open if and only if \(O^c\) is closed. Likewise, a set \(F\) is closed if and only if \(F^c\) is open.
- 3.3 Compact Sets
- Compact Sets
- (Theorem 3.3.4) [Characterization of Compactness in R] A set \(K \in \mathbf{R}\) is compact if and only if it is closed and bounded.
- (Theorem 3.3.5) [Nested Compact Set Property] If \( K_1 \supseteq K_2 \supseteq K_3 \supseteq K_4 \supseteq ...\) is a nested sequence of nonempty compact sets, then the intersection \(\bigcap_{n=1}^{\infty} K_n\) is not empty.
- Open Cover
- 3.4 Perfect Sets and Connected Sets
- 4.2 Functional Limits
- Definitions
- Theorem [4.2.3] Sequential Criterion for Functional Limits
- Corollary [4.2.4] Algebraic Limit Theorem for Functional Limits
- Corollary [4.2.5] Divergence Criterion for Functional Limits
- Example 1: Prove that \(\lim\limits_{x \rightarrow 2} 3x + 1 = 7\)
- Example 2: Prove that \(\lim\limits_{x \rightarrow 2} x^2 = 4\)
- Prove that \(\lim\limits_{x \rightarrow 2} x^2 = 4\)
- Prove that \(\lim\limits_{x \rightarrow 0} x^3 = 0\)
- Construct a rigorous definition in the *challenge-response* style for a limit statement of the form \(\lim\limits_{x \rightarrow c} f(x) = \infty\) and ...
- Construct a definition for the statement of the form \(\lim\limits_{x \rightarrow \infty} f(x) = L\) and use it to prove the \(\lim\limits_{x \rightarrow \infty} 1/x = 0\)
Exercises