Real Variables Midterm 1 Complete Study Guide

Midterm 1 Overview and Format

• Date and Duration: Wednesday, October 7, held in class, with a total duration of 100 minutes.

• Exam Structure and Content Categories:

  1. Definitions

  2. Examples

  3. True / False Questions

  4. Proofs

  5. Proofs

  6. Determining $A^\circ$, $A'$, and $\bar{A}$

  7. Cesàro–Stolz Theorem

  8. Increasing/Decreasing and Bounded Sequences

  9. Metric space proofs and verification

  10. Limit proofs using the $\epsilon, n_\epsilon$ definition

Set Theory Definitions and Properties

• Context: Let $A, B \subseteq X$.

• Core Set Operations:

  • Union: $A \cup B = \{x \in X \mid x \in A \text{ or } x \in B\}$

  • Intersection: $A \cap B = \{x \in X \mid x \in A \text{ and } x \in B\}$

  • Complement: $CA = A^c = \{x \in X \mid x \notin A\}$

  • Set Difference: $A \setminus B = \{x \in A \text{ and } x \notin B\}$

• Fundamental Inclusion Properties:

  1. $A \subseteq A \cup B$ and $B \subseteq A \cup B$

  2. $A \cap B \subseteq A$ and $A \cap B \subseteq B$

• De Morgan's Laws:

  1. $(A \cup B)^c = A^c \cap B^c$

  2. $(A \cap B)^c = A^c \cup B^c$

Order Relations and Real Number Properties

• Definition of Order Relation: A binary relation $R$ on a set $X$ is an order relation if it satisfies three condition:

  1. Reflexivity: $x R x$ for all $x \in X$

  2. Antisymmetry: If $x R y$ and $y R x$, then $x = y$

  3. Transitivity: If $x R y$ and $y R z$, then $x R z$

• Upper and Lower Bounds:

  • Upper Bound: $\alpha$ is an upper bound for $A \subseteq X$ if $\forall x \in A$, $x \le \alpha$.

  • Lower Bound: $\beta$ is a lower bound for $A \subseteq X$ if $\forall x \in A$, $\beta \le x$.

  • Supremum (Least Upper Bound): $\alpha = \sup(A)$.

  • Infimum (Greatest Lower Bound): $\beta = \inf(A)$.

• Precise Characterization of Supremum and Infimum in $\mathbb{R}$:

  • $\alpha = \sup(A)$ if and only if:

  1. $\forall x \in A$, $x \le \alpha$

  2. $\forall \epsilon > 0$, $\exists x_\epsilon \in A$ such that $x_\epsilon > \alpha - \epsilon$

  • $\beta = \inf(A)$ if and only if:

  1. $\forall x \in A$, $x \ge \beta$

  2. $\forall \epsilon > 0$, $\exists x_\epsilon \in A$ such that $x_\epsilon < \beta + \epsilon$

• Archimedean Property (A.P.):

  • General Form: $\forall x > 0$ and $\forall y \in \mathbb{R}$, $\exists n \in \mathbb{N} \setminus \{0\}$ such that $nx > y$

  • Special Case ($x = 1$): $\forall y \in \mathbb{R}$, $\exists n \in \mathbb{N} \setminus \{0\}$ such that $n > y$

  • Special Case (Reciprocal): $\forall x > 0$, $\exists n \in \mathbb{N} \setminus \{0\}$ such that $x > \frac{1}{n}$

  • Integer Interval Corollary: $\forall y \in \mathbb{R}$, $\exists n \in \mathbb{Z}$ such that $n \le y < n + 1$

• Density Properties:

  • Rational Density: $\mathbb{Q}$ is dense in $\mathbb{R}$, meaning $\forall x, y \in \mathbb{R}$ with $x < y$, $\exists z \in \mathbb{Q}$ such that $x < z < y$

  • Irrational Density: $\mathbb{I}$ is dense in $\mathbb{R}$, meaning $forall x, y \in \mathbb{R}$with$x < y$,$\exists s \in \mathbb{I}$such that$x < s < y (requires proof)\n\n\n# Cardinality\n\n* Definitions:\n * Cardinality Equivalence: \text{card}(A) = {B \subset \mathbb{R} \mid \exists \text{ bijection } f : B \rightarrow A}\n * Initial Segment of Natural Numbers: \mathbb{N}n = {1, 2, \dots, n}$with$\text{card}(\mathbb{N}_n) = n\n * Finite Set: A$is finite if$A \sim \mathbb{N}_n$(there exists a bijective mapping$f : A \rightarrow \mathbb{N}_n)\n * Countable Set: \text{card}(\mathbb{N}) = \aleph_0$.$A$is countable if$A \sim \mathbb{N}$(there exists a bijective mapping$f : A \rightarrow \mathbb{N})\n * Uncountable Set: A$is uncountable if$A is neither finite nor countable\n* Examples of Cardinalities:\n * Countable Sets: \mathbb{N}$,$\mathbb{Q}$,$\mathbb{Z}$,$\mathbb{N} \times \mathbb{N}$, and sequence spaces.$\text{card}(\mathbb{N} \times \mathbb{N}) = \aleph_0\n * Uncountable Sets: Real intervals [a, b]$,$\mathbb{R}$, irrational numbers$\mathbb{I}$, complex numbers$\mathbb{C}$, and the Cantor set.$\text{card}(\mathbb{R}) = \mathfrak{c}\n* Cardinality Properties:\n 1. Every infinite set contains a countable subset.\n 2. If A$and$B$are countable, then their Cartesian product$A \times B is countable.\n 3. If A_i$is countable for each$i \in \mathbb{N}$, then the countable union$\bigcup{i=1}^{\infty} A_i is countable.\n 4. Cardinality Hierarchy: \mathbb{N}n \subset \mathbb{N} \subset \mathbb{R} \implies n < \aleph_0 < \mathfrak{c}\n 5. Monotonicity of Cardinality: If A \subseteq B \implies \text{card}(A) \le \text{card}(B)\n\n\n# Metric Spaces\n\n* Metric Definition: A function d : X \times X \rightarrow \mathbb{R}^+$is a metric on$X if it satisfies:\n 1. Symmetry: d(x, y) = d(y, x)\n 2. Identity of Indiscernibles: d(x, y) = 0 \iff x = y\n 3. Triangle Inequality: d(x, z) \le d(x, y) + d(y, z)\n* Examples of Metric Spaces:\n * Standard Metric in \mathbb{R}$:$d(x, y) = |x - y|\n * Metrics in \mathbb{R}^n:\n * Euclidean Metric: d(x, y) = \sqrt{\sum{i=1}^{n} (x_i - y_i)^2}\n * Maximum Metric: d(x, y) = \max_{1 \le i \le n} |x_i - y_i|\n * Sum Metric (Taxicab Metric): d(x, y) = \sum_{i=1}^{n} |x_i - y_i|\n * Discrete Metric: d(x, y) = 1$if$x eq y$, and$d(x, y) = 0$if$x = y\n* Open and Closed Balls:\n * Open Ball: B(x, r) = {y \in X \mid d(x, y) < r}\n * Closed Ball: S(x, r) = {y \in X \mid d(x, y) \le r}\n\n\n# Open, Closed, Interior, Closure, and Limit Points\n\n* Open and Closed Sets:\n * A \subseteq (X, d)$is open if$\forall x \in A$,$\exists B(x, r) \subseteq A\n * A \subseteq (X, d)$is closed if its complement$A^c = X \setminus A is open\n * Examples of Open Sets: (a, b)$,$(0, \infty)$,$\mathbb{R}\n * Examples of Closed Sets: [a, b]$,$[a, \infty)$,$\mathbb{R}, finite sets\n* Fundamental Structural Properties:\n 1. Any finite set is closed (requires proof).\n 2. Arbitrary Union of Open Sets: If A_i$is open, then$\bigcup_{i=1}^{\infty} A_i is open.\n 3. Finite Intersection of Open Sets: \bigcap_{i=1}^{n} A_i is open.\n 4. Arbitrary Intersection of Closed Sets: If B_i$is closed, then$\bigcap_{i=1}^{\infty} B_i is closed.\n 5. Finite Union of Closed Sets: \bigcup_{i=1}^{n} B_i is closed.\n* Interior Points:\n * Definition: x \in A$is an interior point if$\exists B(x, r) \subseteq A.\n * Properties of Interior A^\circ:\n * A^\circ \subseteq A\n * A \subseteq B \implies A^\circ \subseteq B^\circ\n * A^\circ is an open set\n * A$is open$\iff A = A^\circ\n * Examples of Interior:\n * (a, b]^\circ = (a, b)\n * \mathbb{Q}^\circ = \emptyset\n * \mathbb{R}^\circ = \mathbb{R}\n * \mathbb{N}^\circ = \emptyset\n* Closure Points:\n * Definition: x \in X$is a closure point if for any$r > 0$,$B(x, r) \cap A eq \emptyset.\n * Properties of Closure \bar{A}:\n * A \subseteq \bar{A}\n * A \subseteq B \implies \bar{A} \subseteq \bar{B}\n * \bar{A} is a closed set\n * A$is closed$\iff A = \bar{A}\n * Examples of Closure:\n * \overline{(a, b)} = [a, b]\n * \bar{\mathbb{Q}} = \mathbb{R}\n * \bar{\mathbb{I}} = \mathbb{R}\n * \bar{\mathbb{R}} = \mathbb{R}\n * Complement Identities:\n * (A^\circ)^c = \overline{A^c}\n * (\bar{A})^c = (A^c)^\circ\n* Limit Points:\n * Definition: x \in X$is a limit point if for any$r > 0$,$(B(x, r) \setminus {x}) \cap A eq \emptyset\n * Properties of Limit Sets A':\n * A' \subseteq A\n * A \subseteq B \implies A' \subseteq B'\n * Fundamental Identity (PROOF): \bar{A} = A \cup A'\n * Infinitude Property: x \in A' \iff$every neighborhood of$x$contains infinitely many points of$A\n * Finite Set Property: A$finite$\implies A' = \emptyset\n * Closed Set Criterion: A$closed$\iff A' \subseteq A\n * Examples of Limit Sets:\n * \mathbb{Q}' = \mathbb{R}\n * \mathbb{N}' = \emptyset\n * (a, b)' = [a, b]\n * Sets with no limit points: Any finite set\n* Comprehensive Worked Example:\n * Target Set: A = {1 + \frac{1}{n} \mid n \ge 1} = {2, 1 + \frac{1}{2}, 1 + \frac{1}{3}, \dots}\n * Step 1: Finding A^\circ\n * Let \frac{3}{2} \in A$. There is no open ball$(\frac{3}{2} - \epsilon, \frac{3}{2} + \epsilon) \subset A$because every real interval contains irrational numbers which are not in$A$. Thus$A^\circ = \emptyset.\n * Alternative Proof: A \subset \mathbb{Q} \implies A^\circ \subseteq \mathbb{Q}^\circ = \emptyset \implies A^\circ = \emptyset\n * Step 2: Finding A'\n * Sequence limit: \lim_{n \rightarrow \infty} (1 + \frac{1}{n}) = 1\n * Claim: A' = {1}\n * Proof of Claim: Any punctured neighborhood ((1 - \epsilon, 1 + \epsilon) \setminus {1}) \cap A eq \emptyset \iff$there exists$1 + \frac{1}{n}$such that$1 + \frac{1}{n} \in (1 - \epsilon, 1 + \epsilon) \iff$there exists$\frac{1}{n}$such that$-\epsilon < \frac{1}{n} < \epsilon. This is strictly guaranteed by the Archimedean Property.\n * Step 3: Finding \bar{A}\n * \bar{A} = A \cup A' = {1 + \frac{1}{n} \mid n \ge 1} \cup {1}\n\n\n# Bounded Sets\n\n* Definition: A set A \subseteq (X, d) is bounded if its diameter is finite:\n\delta(A) = \sup_{x,y \in A} d(x, y) < \infty\n* Ball Inclusion Criterion: A$is bounded if and only if$A \subseteq B(x, r)$for some$x \in X$and$r > 0.\n\n\n# Compact Sets\n\n* Definition: A set A$is compact if every open cover of$A has a finite subcover:\nA \subseteq \bigcup D_i \implies A \subseteq \bigcup_{i=1}^{n} D_i\n* Fundamental Properties:\n 1. Any finite set is compact (requires proof).\n 2. A$compact$\implies A$is closed and bounded in$(X, d).\n 3. Any closed subset of a compact space (X, d) is compact.\n 4. Heine–Borel Theorem: In \mathbb{R}^n$,$A$is compact$\iff A is closed and bounded.\n* Examples:\n * Closed bounded interval [a, b] is compact.\n * Finite set {1, 2} is compact.\n * Properties of \mathbb{R}$:$\mathbb{R} is open, closed, unbounded, not compact, connected, complete, and uncountable.\n\n\n# Connected Sets\n\n* Definition for Space (X, d)$:$(X, d)$is connected if there are no non-empty open/closed sets$D_1, D_2$such that$X = D_1 \cup D_2$and$D_1 \cap D_2 = \emptyset\n* Definition for Subset A \subseteq (X, d)$:$A$is connected if there are no open/closed sets$D_1, D_2 satisfying:\n 1. A \subseteq D_1 \cup D_2\n 2. A \cap D_1 eq \emptyset\n 3. A \cap D_2 eq \emptyset\n 4. A \cap D_1 \cap D_2 = \emptyset\n* Properties:\n * In \mathbb{R}$,$A$is connected$\iff A is an interval.\n * Examples: (a, b)$is connected;$(1, 2) \cup (3, 4) is not connected.\n * Closure Property: If A$is connected, then$\bar{A} is connected.\n\n\n# Sequences and Convergence\n\n* Convergence Definition in Metric Spaces: x_n \rightarrow x_0 \iff \forall \epsilon > 0, \exists n_\epsilon$such that$\forall n \ge n_\epsilon, d(x_n, x_0) < \epsilon\n* Convergence Definition in \mathbb{R}$:$x_n \rightarrow x_0 \iff \forall \epsilon > 0, \exists n_\epsilon$such that$\forall n \ge n_\epsilon, |x_n - x_0| < \epsilon\n* General Properties:\n 1. x_n$convergent$\implies x_n$bounded ($m \le x_n \le M).\n 2. Limits in metric spaces are unique.\n 3. Sequential Closure Characterization (PROOF): x \in \bar{A} \iff \exists (x_n) \subseteq A$such that$x_n \rightarrow x\n* Monotone Sequences:\n * Definitions:\n * Increasing: x_{n+1} \ge x_n$, which is equivalent to$x_{n+1} - x_n \ge 0\n * Decreasing: x_{n+1} \le x_n$, which is equivalent to$x_{n+1} - x_n \le 0\n * Monotonicity via Ratios (for x_n > 0):\n * If \frac{x_{n+1}}{x_n} > 1 \implies x_n is increasing\n * If \frac{x_{n+1}}{x_n} < 1 \implies x_n is decreasing\n * Monotone Convergence Theorems:\n 1. Monotone Convergence for Supremum (PROOF): If x_n$is increasing and bounded above$\implies \lim_{n \rightarrow \infty} x_n = \sup(x_n)\n 2. Monotone Convergence for Infimum: If x_n$is decreasing and bounded below$\implies \lim_{n \rightarrow \infty} x_n = \inf(x_n)\n 3. Unbounded Increasing Sequence: If x_n$is increasing and unbounded above$\implies x_n \rightarrow \infty\n 4. Unbounded Decreasing Sequence: If x_n$is decreasing and unbounded below$\implies x_n \rightarrow -\infty\n\n\n# Cesàro–Stolz Theorem\n\n* Theorem Statement: If b_n$is strictly increasing and$b_n \rightarrow \infty, then:\n\lim_{n \rightarrow \infty} \frac{a_{n+1} - a_n}{b_{n+1} - b_n} = L \implies \lim_{n \rightarrow \infty} \frac{a_n}{b_n} = L\n* Worked Problem Example:\n * Compute: \lim_{n \rightarrow \infty} \frac{\frac{1}{\ln(2)} + \dots + \frac{1}{\ln(n)}}{n}\n * Set a_n = \frac{1}{\ln(2)} + \dots + \frac{1}{\ln(n)}$and$b_n = n.\n * Note that b_n = n$is strictly increasing and$b_n \rightarrow \infty.\n * Apply difference ratio:\n\lim_{n \rightarrow \infty} \frac{\left(\frac{1}{\ln(2)} + \dots + \frac{1}{\ln(n+1)}\right) - \left(\frac{1}{\ln(2)} + \dots + \frac{1}{\ln(n)}\right)}{(n + 1) - n} = \lim_{n \rightarrow \infty} \frac{1}{\ln(n+1)} = 0\n * Conclusion: By Cesàro–Stolz Theorem, \lim_{n \rightarrow \infty} \frac{\frac{1}{\ln(2)} + \dots + \frac{1}{\ln(n)}}{n} = 0\n\n\n# Epsilon-Limit Proof Techniques\n\n* Worked Problem Example: Show that \lim_{n \rightarrow \infty} \frac{2n + 1}{3n + 5} = \frac{2}{3}.\n* Goal: Show that for any \epsilon > 0$, there exists$n_\epsilon$such that for all$n > n_\epsilon,\n\left|\frac{2n + 1}{3n + 5} - \frac{2}{3}\right| < \epsilon\n* Step-by-Step Algebraic Simplification:\n\left|\frac{2n + 1}{3n + 5} - \frac{2}{3}\right| = \left|\frac{6n + 3 - 6n - 10}{3(3n + 5)}\right| = \left|\frac{-7}{3(3n + 5)}\right| = \frac{7}{3(3n + 5)}\n* Setting up inequality to solve for n:\n\frac{7}{3(3n + 5)} < \epsilon\n3(3n + 5) > \frac{7}{\epsilon}\n9n + 15 > \frac{7}{\epsilon}\nn > \frac{1}{3} \left(\frac{7}{3\epsilon} - 5\right)\n* Selection of threshold n_\epsilon$: Set$n_\epsilon = \tilde{n}$, where$n > \tilde{n} > \frac{1}{3} \left(\frac{7}{3\epsilon} - 5\right), guaranteed to exist by the Archimedean Property.\n\n\n# Metric Verification Methods\n\n* Verification Procedure: To confirm a candidate function d$is a metric on a set$X, rigorously test all 3 required definition parts:\n 1. Verify Symmetry: Check that d(x, y) = d(y, x)$for all$x, y \in X\n 2. Verify Identity of Indiscernibles: Check that d(x, y) = 0 \iff x = y\n 3. Verify Triangle Inequality: Check that d(x, z) \le d(x, y) + d(y, z)$for all$x, y, z \in X$$