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S. Abbott’s Understanding Analysis

1.2: Some Preliminaries

Sets

$$A^c = \{x \in \mathbb{R}: x \not\in A \}$$

De Morgan’s Laws

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

Absolute Value

$$|x| = \begin{cases} x & \text{ if } x \ge 0 \\ -x & \text{ if } x < 0 \\ \end{cases}$$

Satisfies following
i.
$$|ab| = |a | |b|$$

Triangle Inequality

ii. The Triangle Inequality
$$|a+b| \le |a| + |b|$$

Theorem 1.2.6

$a \in \mathbb{R}$ and $b \in \mathbb{R}$ are equal $\iff (|a - b| < \epsilon)$ for all real $\epsilon > 0$

1.3: Axiom of Completeness

Properties of $\ \mathbb{R}$

• Every element of $\mathbb{R}$ has an additive inverse, and every nonzero element has a multiplicative inverse
• $\mathbb{R}$ is a field
• addition and multiplication of real numbers is commutative, distributive, and associative
• Ordering properties on $\mathbb{Q}$ extend to $\mathbb{R}$
• $<, >, \le, \ge$
• $\mathbb{Q} \subset \mathbb{R}$

Axiom of Completeness

Every nonempty set of real numbers that is bounded above has a least upper bound

• Note that this does not apply to $\mathbb{Q}$

Least Upper Bounds and Greatest Lower Bounds

• A set $A \subseteq \mathbb{R}$ is bounded above if $$\exists b \in \mathbb{R} \ \forall a \in A \ \ a \le b$$

• $b$ is an upper bound
• A set $A \subseteq \mathbb{R}$ is bounded below if $$\exists l \in \mathbb{R} \ \forall b \in A \ \ l \le b$$

• $l$ is a lower bound
• A real number $s$ is a Least Upper Bound (or supremum) of $A \subseteq \mathbb{R}$ if the following is true

1. $s$ is an upper bound of $A$
2. if $b$ is any upper bound for $A$, then $s \le b$
• A real number $t$ is a Greatest Lower Bound (or infimum) of $A \subseteq \mathbb{R}$ if the following is true

1. $s$ is a lower bound of $A$
2. if $l$ is any lower bound for $A$, then $l \le t$
• There can only be one supremum (and one infimum)

• The supremum and infimum don’t have to be an element of $S$

• A real number $a_0$ is a maximum of $A$ if $a_0 \in A$ and $\forall a \in A \ \ a_0 \ge a$

• A real number $a_1$ is a minimum of $A$ if $a_1 \in A$ and $\forall a \in A \ \ a_1 \le a$

• There supremum can exist without a maximum, but a maximum always implies the existence of a supremum

Lemma 1.3.8

Suppose $s \in \mathbb{R}$ is an upper bound for a set $A \subseteq \mathbb{R}$.

$s = sup \ A$ if and only if for every $\epsilon > 0$, $\exists a \in A$ satisfying $s - \epsilon < a$

Proof

Another rephrasing of the lemma: If $s$ is an upper bound, $s$ is the least upper bound if and only if any number smaller than $s$ is not an upper bound.

Use this to prove the forward and reverse implications.

2.2: The Limit of a Sequence

Definition 2.2.3 (Convergence of a Sequence)

A sequence $(a_n)$ converges to $a \in \mathbb{R}$ if
$$\forall \epsilon > 0\ \exists N \in \mathbb{N} \ni (n \ge N \implies |a_n - a| < \epsilon)$$

4.2: Functional Limits

Definition 4.2.1 (Functional Limit)

Let $f: A \to \mathbb{R}$ and let $c \in A$. Let $x \in A$
We say $\lim_{x\to c} f(x) = L$ if

$$\forall \epsilon > 0, \exists \delta > 0 \ni (0 < |x-c| < \delta \implies| f(x) - L| < \epsilon)$$

• Note that $0 < |x-c|$ is just a compact way of saying $x \neq c$

Theorem 4.2.3 (Sequential Criterion for Functional Limits)

Given $f: A \to \mathbb{R}$ and a limit point $c \in A$, the following are equivalent
i. $\lim\limits_{x\to c} f(x) = L$
ii. $\forall (x_n) \subseteq A$ satisfying $x_n \neq c$, it follows that $f(x_n) \to L$

Corollary 4.2.5 (Divergence Criterion for Functional Limits)

Let $f: A \to \mathbb{R}$ and let $c$ be a limit point of $A$

If $\exists (x_n), (y_n) \subseteq A \ni x_n \neq c \land y_n \neq c$ and
$$\lim x_n = \lim y_n = c \text{ but } \lim f(x_n) \neq \lim f(y_n)$$
then $\lim_{x\to c} f(x) = DNE$

4.3: Continuous Functions

Definition 4.3.1 (Continuity)

A function $f: A \to \mathbb{R}$ is continous at a point $c \in A$ if

$$\forall \epsilon > 0, \exists \delta > 0 \ni (x \in A \land |x - c| < \delta \implies |f(x) - f(c)| < \epsilon )$$

Theorem 4.3.2 ( Characterization of Continuity)

Let $f: A \to \mathbb{R}$ and $c\in A$. Let $x\in A$.

$f$ is continuous is equivalent to saying any of the three following equivalent statements
i.
$$\forall \epsilon > 0, \exists \delta > 0 \ni |x-c| < \delta \implies |f(x) - f(c) | < \epsilon$$
ii.
$$\forall V_\epsilon (f(c)), \exists V_\delta (c) \ni \left(x \in V_\delta (c) \implies f(x) \in V_\epsilon(f(c))\right)$$
iii.
$$\forall (x_n) \subseteq A \to c \implies f(x_n) \to f(c)$$

iv. If $c$ is a limit point of $A$
$$\lim_{x\to c} f(x) = f(c)$$

Corollary 4.3.3 (Criterion for Discontinuity)

Let $f: A \to \mathbb{R}$ and let $c \in A$ be a limit point of $A$

If $\exists (x_n) \subseteq A$ where $(x_n) \to c$ but $f(x_n)$ does not converge to $f(c)$, then $f$ is not continous at $c$

Theorem 4.3.4 (Algebraic Continuity Theorem)

Assume $f: A \to \mathbb{R}$ and $g: A \to \mathbb{R}$ are continous at $c \in A$. Then the following are all continous at $c \in A$

i. $kf(x) \forall k \in \mathbb{R}$
ii. $f(x) + g(x)$
iii. $f(x) g(x)$
iv. $f(x) / g(x)$ if $g(c) \neq 0$

4.4: Continous Functions on Compact Sets

Theorem 4.4.1 (Preservation of Compact Sets)

Let $f : A \to \mathbb{R}$ be continous on $A$.

If $K \subseteq A$ is compact, then $f(K)$ is compact as well

Theorem 4.4.2: Extreme Value Theorem

If $f: K \to \mathbb{R}$ is continous on a compact set $K \subseteq \mathbb{R}$, then $f$ attains a maximum and minimum value.

$$\exists x_0, x_1 \in K \ni f(x_0) \le f(x) \le f(x_1)\ \forall x \in K$$

Definition 4.4.4: Uniform Continuity

$f: A \to \mathbb{R}$ is uniformly continous on $A$ if

$$\forall \epsilon > 0 \ \exists \delta > 0, \ni \forall x, y \in A \ (|x-y| < \delta \implies |f(x) - f(y)| < \epsilon)$$

• Note that a function is continous if $\forall c \in A \ \forall \epsilon > 0, \exists \delta > 0 \ni |x-c| < \delta \implies |f(x) - f(c) | < \epsilon$
• For Regular continuity $\delta$ could depend on the value of $c$, but for uniform continuity, $\delta$ does not depend on the $c \in A$

Sequential Criterion for Absence of Uniform Continuity

A function $f: A \to \mathbb{R}$ fails to be uniformly continous on $A$ if and only if
$\exists \epsilon_0 > 0$ and $\exists (x_n), (y_n) \subseteq A$ such that
$$|x_n - y_n| \to 0\text{ but } |f(x_n) - f(y_n)| \ge \epsilon_0$$

Theorem 4.4.7 (Uniform Continuity on Compact Sets)

A function is that is continous on a compact set $K$ is uniformly continous on $K$

4.5: The Intermediate Value Theorem

Theorem 4.5.1 (Intermediate Value Theorem)

Let $f: [a, b] \to \mathbb{R}$ be continuous. If $L \in \mathbb{R}$ and $f(a) < L < f(b)$ or $f(b) > L > f(a)$, then $\exists c \in (a, b)$ where $f(c) = L$

Theorem 4.5.2 (Preservation of Connected Sets)

Let $f: G \to \mathbb{R}$ be continuous. If $E \subseteq G$ is connected, then $f(E)$ is connected.

5.2: Derivatives

Definition 5.2.1 (Differentiability)

Let $g: A \to \mathbb{R}$ be defined on interval $A$. Given $c \in A$, the derivative of $g$ at $c$ is given by
$$g’(c) = \lim_{x\to c} \frac{g(x) - g(c) }{x-c}$$
if the limit exists

Theorem 5.2.3

If $g: A\to \mathbb{R}$ is differentiable at $c \in A$, then $g$ is continuous at $c$

Theorem 5.2.4 (Algebraic Differentiability Theorem)

Let $f$ and $g$ be defined on interval $A$

Theorem 5.2.6 (Interior Extermum Theorem)

Let $f$ be differentiable on $(a, b)$. If $f$ attains a maximum value at $c \in (a, b)$, then $f’(c) = 0$. The same applies if $f$ attains a minimum.

Theorem 5.2.7 (Darboux’s Theorem)

If $f$ is differentiable on $[a, b]$ and if $f’(\alpha) < \alpha < f’(b)$, then $\exists c \in (a, b) \ni f’(c) = \alpha$

5.3: The Mean Value Theorem

Theorem 5.3.1 (Rolle’s Theorem)

Let $f: [a, b] \to \mathbb{R}$ be continuous on $[a, b]$ and differentiable on $(a, b)$. If $f(a) = f(b)$, $\exists c \in (a, b)$ where $f’(c) = 0$

Theorem 5.3.2 (Mean Value Theorem)

Let $f: [a, b] \to \mathbb{R}$ be continuous on $[a, b]$ and differentiable on $(a, b)$, then there exists $c \in (a, b)$ where $f’(c) = \frac{f(b) - f(a)}{b-a}$

Theorem 5.3.6: L’Hospital’s Rule

$$\lim_{x \to a} \frac{f’(x)}{g’(x)} = L \implies \lim_{x\to a} \frac{f(x)}{g(x)} = L$$

6.2: Uniform Convergence of a Sequence of Functions

Definition 6.2.1: Pointwise Convergence

Let $f_n$ be a function defined for each $n \in \mathbb{R}$ on set $A\subseteq \mathbb{R}$

The sequence $(f_n)$ converges pointwise on $A$ to $f$ if
$$\forall x \in A, f_n(x) \to f(x)$$

Definition 6.2.3 (Uniform Convergence)

Let $(f_n)$ be a sequence of functions defined on $A \subseteq \mathbb{R}$. Then $(f_n)$ converges uniformly on $A$ to $f$ on $A$ if
$$\forall \epsilon > 0, \exists N \in \mathbb{N} \ni$$
$$n\ge N \land x \in A \implies |f_n(x) - f(x)| < \epsilon$$

Note the definiton of pointwise convergence:
$$\forall x \in A \ \forall \epsilon > 0, \exists N \in \mathbb{N} \ni$$
$$n \ge N \implies | f_n(x) - f(x) | < \epsilon$$

• Similar to uniform continuity, the value for $N$ does not depend on $x$ just like how $\delta$ did not depend on the value of $x$

Cauchy Criterion for Uniform Convergence

$(f_n)$ defined on $A$ converges uniformly on $A$ if and only if
$$\forall \epsilon > 0, \exists N \in \mathbb{N} \ni$$
$$m, n \ge N \land x \in A \implies |f_n(x) - f_m(x) | < \epsilon$$

Theorem 6.2.6 (Continuous Limit Theorem)

Let $(f_n)$ be a sequence of functions on $A$ that converge uniformly on $A$ to $f$. If each $f_n$ is continuous at $c \in A$, then $f$ is continous at $c$

6.4: Series of Functions

Definition 6.4.1

For each $n\in \mathbb{N}$, let $f_n$ and $f$ be functions on $A$.

Then the infinite series
$$\sum_{n=1}^\infty f_n(x) = f_1(x) + f_2(x) + f_3(x) + … f_n(x) + …$$
converges pointwise on $A$ to $f(x)$ if the sequence of partial sums $s_k(x)$ converge pointwise to $f(x)$. If $s_k$ converges uniformly on $A$, then the infinite series converges uniformly on $A$

$$s_k(x) = f_1(x) + f_2(x) + … + f_k(x)$$

Definition 6.4.2 (Term-by-term Continuity Theorem)

Let $f_n$ be continuous functions defined on a set $A \subseteq \mathbb{R}$ and assume $\sum_{n=1}^\infty f_n$ converges uniformly on $A$ to $f$. Then $f$ is continuous on $A$

Theorem 6.4.4 (Cauchy Criterion for Uniform Convergence of Series)

A series $\sum_{n=1}^\infty f_n$ converges uniformly on $A \subseteq \mathbb{R}$ if and only if
$$\forall \epsilon > 0 \exists N \in \mathbb{N} \ni$$
$$|f_{m+1} (x) + f_{m+2}(x) + f_{m+3}(x) + … + f_n(x) < \epsilon$$
whenever $n > m\ge N$ and $x \in A$

Corollary 6.4.5 (Weierstrauss M-Test)

For each $n\in \mathbb{N}$, let $f_n$ be defined on $A$ and let $M_n > 0$ be a real number satisfying

$$|f_n(x)| \le M_n \ \forall x \in A$$

If $\sum_{n=1}^\infty M_n$ converges, then $\sum_{n=1}^\infty f_n$ converges uniformly on $A$