How We Got From There To Here: A Story of Real Analysis

orthogonality of, Problem

Taylor’s series for, Problem

divergence to, Definition

negative infinity

positive infinity

first series expansion, Problem

second series expansion, Problem

\(\QQ\) is countable, Problem

creating irrationals from rationals, Problem

creating rationals from irrationals, Problem

has measure zero in \(\RR\), Problem

is countable, Theorem

is it possible to have two irrational numbers, \(a\) and \(b\text{,}\) such that \(a^b\) is rational, Problem

is it possible to have two rational numbers, \(a\) and \(b\text{,}\) such that \(a^b\) is irrational, Problem

rational numbers exist between rational numbers, Problem

sums and products of rational and irrational numbers, Problem

addition of Cauchy sequences, Problem

any complete, linearly ordered field is isomorphic to, Theorem

as Cauchy sequences

defined by Cauchy sequences, Problem

defining infinite decimal addition, Problem

irrational numbers, Problem Problem

irrational numbers drill, 5 parts, Problem

is uncountable

ordering Dedekind cuts, Problem

products of rationals and irrationals, Problem

real numbers exist between real numbers, Theorem

the number \(1\) as a Cauchy sequence, Problem

as a power series, Problem

derivative of series form, Problem

is continuous for \(0\leq x\lt \frac{\pi}{2}\), Problem

orthogonality of, Problem

Taylor’s series for, Problem

value of, Problem

is irrational, Paragraph

meaning of, Paragraph

is continuous at every positive real number, Problem

is continuous at zero, Problem

\(e^{a+b}=e^ae^b\), Problem

as the solution of an Initial Value Problem, Example

definition of \(e\), Definition

Taylor’s series for, Problem

converges pointwise on \([0,1]\), Problem

converges uniformly on \((0,b),\) \(b\lt 1\), Problem

Abel’s Lemma, Problem

Abel’s Partial Summation Formula, Problem

Abel’s Theorem, Theorem Problem

and \(\QQ\), Problem

and LUBP, Problem

"Tanquam ex ungue leonem", Paragraph

Bernoulli’s challenge, Paragraph

portrait of, Figure

\(g(c)=\frac{c-x}{1+c}\) is increasing, Problem

as a power series centered at zero, Problem

converges on the interval \([0,1]\), Theorem

squaring the, Problem

portrait of, Figure

implies that a continuous functions on a closed set is bounded, Problem

implies the NIP, Problem

Bernoulli’s solution, Paragraph

and the modern view of mathematics, Paragraph

Cantor’s Theorem, Theorem Problem

first proof that \(\RR\) is uncountable, Theorem

fourth theorem on the uniqueness of Fourier series, Theorem

portrait of, Figure

uniqueness of Fourier series

unit interval and unit square have equal cardinalty, Paragraph

countable sets, Definition

equal cardinality, Definition

of a power set, Problem

Cauchy’s counterexample

Cauchy’s flawed proof that the limit of continuous functions is continuous, Problem

portrait of, Figure

\(\pm\sqrt{x}\) is continuous at zero, Problem

\(\sin e^x\) is continuous everywhere, Problem

\(e^x\) is continuous everywhere, Problem

\(f(x) = mx +b\) is continuous everywhere, Problem

Bolzano-Weierstrass Theorem implies a continuous function on a closed set is bounded, Problem

Cauchy’s flawed proof that the limit of continuous functions is continuous, Problem

definition of, Definition

drill problems, Problem

Extreme Value Theorem (EVT) and, Paragraph

formal definition of discontinuity, Problem

Heaviside’s function is not continuous at zero, Problem

implied by differentiability, Theorem

Intermediate Value Theorem and, Paragraph

larger \(\eps\) works in definition, Problem

of \(D(x)= \begin{cases}x,\amp \text{ if } x\text{ is rational } \\ 0,\amp \text{ if } x\text{ is irrational } \end{cases} \), Problem

of \(D(x)= \begin{cases}x,\amp \text{ if } x\text{ is rational } \\ 0,\amp \text{ if } x\text{ is irrational } \end{cases} \), Problem

smaller \(\delta\) works in definition, Problem

smaller \(\delta\text{,}\) bigger \(\eps\), Problem

via limits, Theorem Problem

via sequences, Problem

Weierstrass’s continuous, but non-differentiable function, Problem

\(\ln x\) is continuous everywhere, Problem

\(e^x\) is continuous everywhere, Problem

a constant function is continuous, Problem

continuous function on a closed, bounded interval is bounded, Theorem

if \(f\) is continuouse and \(f(a)\neq0\) then \(f\) is bounded away from zero near a, Problem

on a closed set, and the Bolzano-Weierstrass Theorem, Problem

sum of continuous functions is continuous, Theorem

the composition of continuous functions is continuous, Theorem Problem

the product of continuous functions is continuous, Problem

the quotient of continuous functions is continuous, Problem

uniform convergence and, Theorem

uniform limit of continuous functions is continuous, Theorem

generalized, Conjecture

original, Conjecture

definition of nonconvergence of a sequence, Problem

of a sequence, Definition

of a series

pointwise convergence, Problem

pointwise vs. uniform convergence, Problem

the radius of convergence of a power series, Problem

uniform convergence, Problem

countable sets drill, 5 parts, Problem

countable union of finite sets, Problem

defintion of, Definition

deleting a countable subset, Problem

unions and intersections of, Problem

Dedekind cuts, Paragraph Definition

portrait of, Figure

\(f^\prime(a)>0\) implies \(f\) is increasing nearby, Problem

\(f^\prime(a)\lt 0\) implies \(f\) is decreasing nearby, Problem

definition of the derivative, Definition

differentiability implies continuity, Theorem Problem

differentiation of a sequence of functions, Problem

if \(f^\prime\lt 0\) on an interval then \(f\) is decreasing, Problem

of \(\sin x\) as a series, Problem

of the pointwise limit of functions, Theorem

power rule with fractional exponents, Problem

term by term differentiation of power series, Problem

divergence to infinity implies divergence, Problem

of a sequence, Definition

of a series

Basel Problem, the, Problem

Euler’s constant \((\gamma)\)

Euler’s Formula, Problem

portrait of, Figure

continuity and, Paragraph

Rolle’s Theorem, and, Section

problems leading to

if \(f(a)\) is a maximum then \(f^\prime(a)=0\), Problem

if \(f(a)\) is a minimum then \(f^\prime(a)=0\), Problem

any complete, linearly ordered field is isomorphic to \(\RR\text{.}\), Theorem

Cantor’s first theorem on uniqueness, Theorem

Cantor’s second theorem on uniqueness, Theorem

Cantor’s third theorem on uniqueness, Theorem

computing the coefficients, Problem

cosine series

divergent Fourier series example, Problem

sine series of an odd function, Problem

term by term differentiation of, Problem

uniform convergence and, Problem

portrait of, Figure

definition of, Problem

portrait of, Figure

portrait of, Figure

fundamental solutions of, Problem

parameter \(k\) must be less than zero, Problem

solving for \(\xi(x)\), Problem

a polynomial with odd degree must have a root, Problem

continuity and, Paragraph

the case \(f(a)\geq v\geq f(b)\), Problem

the case \(f(a)\leq v\leq f(b)\), Problem

portrait of, Figure

\(\ln 2\), Problem

\(x\lt a\), Problem

doesn’t hold in \(\QQ\), Problem

identifying suprema and infima, Problem

implies the Archimedean Property, Problem

implies the existence of irrational numbers, Problem

implies the Nested Interval Property (NIP), Problem

and infinitesimals, Paragraph

differentiation rules, Paragraph

first calculus publication, Paragraph

Leibniz’s product rule, Problem

portrait of, Figure

\(\lim_{n\rightarrow\infty}b^n=0\) if \(-1\lt b\lt 1\), Problem

\(\lim_{n\rightarrow\infty}b^{\left(\frac{1}{n}\right)}=1\) if \(b>0\), Problem

\(\limit{x}{0}{\textstyle\frac{\sin x}{x}}=1\), Problem

\(\limit{x}{a}{\frac{x^2-a^2}{x-a}}=2a\), Problem

\(\limit{x}{a}{f(x)}=f(a)\) implies \(f(x)\) is continuous, Problem

accumulation point, Problem

accumulation points, Definition

definition of non-existence, Problem Problem

identifing the theorems used in a limit, Problem

of a constant sequence, Problem

of a constant times a sequence, Problem

of interval endpoints in the NIP, Theorem

of ratios of polynomials, Problem

of the difference of sequences, Problem

products of, Theorem

properties of, Theorem Problem

quotients of, Theorem

Squeeze Theorem, Problem

Squeeze Theorem for Sequences, Theorem

termwise sums of, Theorem

verify limit laws from calculus, Problem

verifying limits via continuity, Problem

\(\QQ\) has measure zero in \(\RR\), Problem

endpoints, Problem Theorem Problem

implies the existence of square roots of integers, Problem

implies the LUBP, Problem

square roots of integers, and, Problem

weak form, Theorem Problem

foundation of calculus, Paragraph

portrait of, Figure

\(\CC\) is a field, Problem

\(\QQ\) is a field, Problem

for Cauchy sequences, Problem

linearly ordered, Definition Problem

of \(\cos nx\), Problem

of \(\sin nx\), Problem

derivative and, Theorem

infinite, Paragraph

with odd degree must have a root, Problem

a power series diverges outside it’s radius of convergence, Problem

converge inside radius of convergence, Theorem

converge uniformly inside their radius of convergence, Problem

converge uniformly on their interval of convergence, Theorem

definition of, Definition

drills, Problem

for \(a^x\) expanded about 0, Problem

of \(\sin(x)\text{,}\) expanded about \(a\), Problem

term by term derivative of, Problem Theorem Problem

term by term integral of, Problem

term by term integration of, Problem

the radius of convergence, Problem Problem

uniform convergence of, Problem

Weierstrass-\(M\) Test and, Problem

first proof, Problem

second proof, Problem

all subsequences of a convergent sequence converge, Problem

bounded and non-decreasing, Problem

Cauchy sequences, Definition

constant multiples of, Problem

constant sequences, Problem

convergence, Definition Theorem Problem

convergence of

convergence to zero, Definition

definition of divergence, Problem

difference of, Problem

differentiation of a sequence of functions, Problem

divergence of, Definition Problem

divergence to \(\infty\), Definition Problem Problem

divergence, but not to infinity, Problem Problem

find a bounded sequence of rational numbers such that no subsequence converges to a rational number, Problem

Geometric, Problem

if the sequence \(\left(s_n\right)\) is bounded then \(\lim_{n\rightarrow\infty}\left(\frac{s_n}{n}\right)=0\), Problem

limit is unique, Problem

lower and upper bounds for, Problem

not subsequences, Example

Ratio Test for, Problem

subsequences, Definition Example

termwise product of, Problem

termwise quotient of, Problem

termwise sums of, Problem Problem

the sequence of positive integers diverges to infinity, Problem

\(\tan^{-1}x\), Problem

\(n\)th term test, Problem

absolute convergence of

Alternating Harmonic Series

Binomial Series, the, Theorem

Cauchy Criterion, Problem

Cauchy sequences

Comparison Test, Theorem

Geometric Sequence

Geometric series, Problem

graph the square root series, Problem

Harmonic Series, Paragraph Paragraph

rearrangements, Theorem

solutions of \(\frac{\dx^2y}{\dx{ x}^2}=-y\), Problem

Taylor’s series, Theorem

term by term integration of, Problem

the Comparison Test, Problem

accumulation points, Definition Problem

cardinality of a power set, Problem

countably infinite subsets, Problem

derived sets, Definition

intervals are uncountable, Problem

power set, Problem

for functions, Theorem Problem

portrait of, Figure

drill problems, Problem

use to obtain the general binomial series, Problem

is continuous at zero, Problem

modified version is not continuous at zero, Problem

for Integrals, Problem

Reverse Triangle Inequalitiy, Item

Triangle Inequality, Item Problem

continuous functions and, Theorem Problem

Fourier Series and, Problem

integration and, Theorem Problem Problem

of power series at the endpoints of the interval of convergence, Problem

positive power series and, Problem

power series and, Problem

continuous, everywhere non-differentiable function, Problem

portrait of, Figure

drill problems, Problem