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S. Abbott’s Understanding Analysis

1.2: Some Preliminaries

Sets

$$A^c=\{x\in \mathbb{R}:x\notin A\}$$

De Morgan’s Laws

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

Absolute Value

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

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 $⟺(|a-b|<ϵ)$ for all real $ϵ>0$

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 $$\mathrm{\exists }b\in \mathbb{R}\mathrm{\forall }a\in Aa\le b$$

  • $b$ is an upper bound

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

  • $l$ 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 $\mathrm{\forall }a\in A{a}_0\ge a$

• A real number $a_1$ is a minimum of $A$ if $a_1\in A$ and $\mathrm{\forall }a\in A{a}_1\le a$

• 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 $A\subseteq \mathbb{R}$.

$s=supA$ if and only if for every $ϵ>0$, $\mathrm{\exists }a\in A$ satisfying $s-ϵ<a$

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
$$\mathrm{\forall }ϵ>0\mathrm{\exists }N\in \mathbb{N}\ni (n\ge N⟹|a_n-a|<ϵ)$$

3.2: Open and Closed Sets

3.3: Compact Sets

3.4: Perfect Sets and Connected Sets

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 $\underset{x\to c}{lim}f(x)=L$ if

$$\mathrm{\forall }ϵ>0,\mathrm{\exists }\delta >0\ni (0<|x-c|<\delta ⟹|f(x)-L|<ϵ)$$

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

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. $\underset{x\to c}{lim}f(x)=L$
ii. $\mathrm{\forall }(x_n)\subseteq A$ satisfying $x_n\ne c$, it follows that $f(x_n)\to L$

Corollary 4.2.4 (Algebraic Limit Theorem for Functional Limits)

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 $\mathrm{\exists }(x_n),(y_n)\subseteq A\ni x_n\ne c\wedge y_n\ne c$ and
$$lim{x}_n=lim{y}_n=c\text{ but }limf(x_n)\ne limf(y_n)$$
then $\underset{x\to c}{lim}f(x)=DNE$

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

$$\mathrm{\forall }ϵ>0,\mathrm{\exists }\delta >0\ni (x\in A\wedge |x-c|<\delta ⟹|f(x)-f(c)|<ϵ)$$

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.
$$\mathrm{\forall }ϵ>0,\mathrm{\exists }\delta >0\ni |x-c|<\delta ⟹|f(x)-f(c)|<ϵ$$
ii.
$$\mathrm{\forall }{V}_{ϵ}(f(c)),\mathrm{\exists }{V}_{\delta }(c)\ni (x\in V_{\delta }(c)⟹f(x)\in V_{ϵ}(f(c)))$$
iii.
$$\mathrm{\forall }(x_n)\subseteq A\to c⟹f(x_n)\to f(c)$$

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

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 $\mathrm{\exists }(x_n)\subseteq A$ 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. $kf(x)\mathrm{\forall }k\in \mathbb{R}$
ii. $f(x)+g(x)$
iii. $f(x)g(x)$
iv. $f(x)/g(x)$ if $g(c)\ne 0$

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.

$$\mathrm{\exists }{x}_0,x_1\in K\ni f(x_0)\le f(x)\le f(x_1)\mathrm{\forall }x\in K$$

Definition 4.4.4: Uniform Continuity

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

$$\mathrm{\forall }ϵ>0\mathrm{\exists }\delta >0,\ni \mathrm{\forall }x,y\in A(|x-y|<\delta ⟹|f(x)-f(y)|<ϵ)$$

• Note that a function is continous if $\mathrm{\forall }c\in A\mathrm{\forall }ϵ>0,\mathrm{\exists }\delta >0\ni |x-c|<\delta ⟹|f(x)-f(c)|<ϵ$
• 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
$\mathrm{\exists }{ϵ}_0>0$ and $\mathrm{\exists }(x_n),(y_n)\subseteq A$ such that
$$|x_n-y_n|\to 0\text{ but }|f(x_n)-f(y_n)|\ge {ϵ}_0$$

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 $\mathrm{\exists }c\in (a,b)$ 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
$$g^{\prime }(c)=\underset{x\to c}{lim}\frac{g(x)-g(c)}{x-c}$$
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 5.2.5 (Chain Rule)

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 $f^{\prime }(c)=0$. The same applies if $f$ attains a minimum.

Theorem 5.2.7 (Darboux’s Theorem)

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

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 $f(a)=f(b)$, $\mathrm{\exists }c\in (a,b)$ where $f^{\prime }(c)=0$

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 $f^{\prime }(c)=\frac{f(b)-f(a)}{b-a}$

Theorem 5.3.6: L’Hospital’s Rule

$$\underset{x\to a}{lim}\frac{f^{\prime }(x)}{g^{\prime }(x)}=L⟹\underset{x\to a}{lim}\frac{f(x)}{g(x)}=L$$

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
$$\mathrm{\forall }x\in A,f_n(x)\to f(x)$$

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
$$\mathrm{\forall }ϵ>0,\mathrm{\exists }N\in \mathbb{N}\ni$$
$$n\ge N\wedge x\in A⟹|f_n(x)-f(x)|<ϵ$$

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

• 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
$$\mathrm{\forall }ϵ>0,\mathrm{\exists }N\in \mathbb{N}\ni$$
$$m,n\ge N\wedge x\in A⟹|f_n(x)-f_m(x)|<ϵ$$

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
$$\sum _{n=1}^{\mathrm{\infty }}{f}_n(x)=f_1(x)+f_2(x)+f_3(x)+\dots f_n(x)+\dots$$
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 $A$

$$s_k(x)=f_1(x)+f_2(x)+\dots +f_k(x)$$

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 $\sum _{n=1}^{\mathrm{\infty }}{f}_n$ 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 $\sum _{n=1}^{\mathrm{\infty }}{f}_n$ converges uniformly on $A\subseteq \mathbb{R}$ if and only if
$$\mathrm{\forall }ϵ>0\mathrm{\exists }N\in \mathbb{N}\ni$$
$$|f_{m+1}(x)+f_{m+2}(x)+f_{m+3}(x)+\dots +f_n(x)<ϵ$$
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

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

If $\sum _{n=1}^{\mathrm{\infty }}{M}_n$ converges, then $\sum _{n=1}^{\mathrm{\infty }}{f}_n$ converges uniformly on $A$

6.5: Power Series