Index Index
Abel, Niels Henrik
Abel’s Lemma Problem
Abel’s Partial Summation Formula Problem
portrait of Figure
Absolute Convergence Definition
absolute value Problem
absolute value of a Dedekind cut Definition
accumulation points
definition Definition
addition of Cauchy sequences Definition
addition of Dedekind cuts Definition Definition
and \(\QQ\) Problem
and LUBP Problem
\(b\) is an upper bound of \(S\subseteq \RR\) if and only if \(-b\) is a lower bound of \(-S\) Problem
Bernoulli
Jacob Paragraph
Johann Paragraph
challenge problem Paragraph
portrait of Figure
Binomial series
squaring the Problem
Binomial series, the Problem
\(g(c)=\frac{c-x}{1+c}\) is increasing Problem
as a power series centered at zero Problem
converges on the interval \([0,1]\) Theorem
portrait of Figure
Bolzano–Weierstrass Theorem Theorem
implies that a continuous functions on a closed set is bounded Problem
implies the NIP Problem
proof of Problem
Johanne Bernoulli’s solution Paragraph
BWT. See Bolzano–Weierstrass Theorem
first proof that \(\RR\) is uncountable Theorem
fourth theorem on the uniqueness of Fourier series Theorem
portrait of Figure
uniqueness of Fourier series
first theorem on Theorem
second theorem Theorem
third theorem on Theorem
unit interval and unit square have equal cardinalty Paragraph
Cantor’s middle–third set
construction of Figure
has measure zero Problem
is uncountably infinite Problem
Cardano, Girolomo
portrait of Figure
cardinality
countable sets Definition
definition Paragraph Definition
of a power set Problem
Cauchy Sequence Definition
Cauchy’s counterexample
part 1 Problem
part 2 Problem
Cauchy’s flawed proof that the limit of continuous functions is continuous Problem
portrait of Figure
Cohen, Paul
portrait of Figure
Completeness Axiom, (the NIP) Axiom
continuity Problem
\(\pm\sqrt{x}\) 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
and sequences Theorem
at a point Definition
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
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 \(\ln{x}\) Problem
of \(D(x)= \begin{cases}x,\amp \text{ if } x\text{ is
rational } \\ 0,\amp \text{ if } x\text{ is irrational }
\end{cases} \) using sequences Problem
of \(D(x)= \begin{cases}x,\amp \text{ if } x\text{ is rational }
\\ 0,\amp \text{ if } x\text{ is irrational } \end{cases}\) using the definition Problem
of \(f(x)=0\) at zero Example
of a constant function Problem
of a product Problem
of a quotient Problem
of a sum Theorem
on an interval Paragraph
smaller \(\delta\) works in definition Problem
uniform Problem
via sequences Problem
Weierstrass’ continuous, but non–differentiable function Problem
continuous functions
\(\ln x\) 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
on a closed, bounded interval, and the Bolzano–Weierstrass Theorem Problem
sum of continuous functions is continuous Theorem
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
Continuum Hypothesis
generalized Conjecture
original Conjecture
convergence
definition of nonconvergence of a sequence Problem
of a sequence
implies the convergence of the absolute sequence Problem
of a series
absolute Definition
absolute convergence implies convergence Problem
Comparison Test Problem
pointwise Definition
pointwise convergence Problem
pointwise vs. uniform convergence Problem
the radius of convergence of a power series Problem
uniform convergence Problem
\(\cos (nx)\)
orthogonality of Problem
\(\cos x\)
Taylor’s series for Problem
countable sets
countable sets drill, 5 parts Problem
countable union of finite sets Problem
definition Definition
defintion of Definition
unions and intersections of Problem
Darboux Integrability Definition
Darboux, Jean
portrait of Figure
Dedekind cut Definition
addition of Definition
multiplication of positive cuts Definition
ordering of Definition
Dedekind cuts as sets Definition
Dedekind cuts Paragraph Definition
closure of Theorem
multiplication of Problem
order properties Problem
ordering of Theorem
subtraction of Problem
technical lemma for Problem
portrait of Figure
Definition
Absolute Convergence Definition
absolute value of a Dedekind cut Definition
accumulation points Definition
addition and multiplication of Cauchy sequences Definition
addition of Dedekind cuts Definition Definition
cardinality Definition
Cauchy Sequence Definition
characteristic function Paragraphs
closed set Paragraph
continuity
at a point Definition
convergence of a sequence Definition
convergence of a sequence to zero Definition
countable sets Definition
Darboux Integrability Definition
Dedekind cut Definition
Dedekind cuts as sets Definition
derivative Definition Definition
derived sets Definition
divergence of a sequence Definition
divergence of a sequence to \(\pm\infty\) Definition
equivalent Cauchy sequences Definition
\(e^x\) Definition
Greatest Lower Bound Property Problem
image Definition (see also pre–image)
infimum Problem
inner measure Definition
Least Upper Bound Definition
Lebesgue integral Definition
Lebesgue measure Definition
Legesgue Integral
upper and lower Definition
limit Definition
linear order Definition
multiplication of positive Dedekind cuts Definition
multiplication of Dedekind cuts Definition
near Definition
open conver Definition
open set Definition
ordering of Dedekind cuts Definition
outer measure Definition
partition refinement Definition
Pointwise Convergence Definition
power series Definition
properties of a measure Definition
real numbers defined as Cauchy sequences Definition
Riemann Integral Definition
simple function Definition
subsequences Definition
subtraction of Dedekind cuts Definition
topology on a set Definition
Uniform Continuity Definition
Uniform Convergence Definition
Upper Bound Definition
definition Definition
differentiation
definition of the derivative Definition Definition
differentiation of a sequence of functions Problem
of \(\sin x\) as a power series Problem
of the pointwise limit of functions Theorem
power rule with fractional exponents Problem
term by term differentiation of power series Problem
Dirichlet, Lejeune
portrait of Figure
Dirichlet’s Function Problem
divergence
divergence to infinity implies divergence Lemma
of a sequence Definition
of a series
\(n\)th term test Problem
divergence of a sequence to \(\pm\infty\)
definition Definition
equivalent Cauchy sequences
definition Definition
Euler, Leonhard Paragraph
Basel Problem, the Problem
Euler’s constant \((\gamma)\)
existence of Problem
slow convergence to Problem
Euler’s Formula Problem
portrait of Figure
\(e^x\)
\(e^{a+b}=e^ae^b\) Problem
as the solution of an Initial Value Problem Example
definition Definition
Taylor’s series for Problem
Extreme Value Theorem Problem
continuity and Paragraph
problems leading to
if a prime divides a product of two numbers then it divides one of the factors Problem
if a prime divides an arbitrary product then it divides one of the factors Problem
fields
any complete, linearly ordered field is isomorphic to \(\RR\text{.}\) Theorem
Fourier Series Problem
Cantor’s first theorem on uniqueness Theorem
Cantor’s second theorem on uniqueness Theorem
Cantor’s third theorem on uniqueness Theorem
computing the coefficients of Problem
cosine series
the Fourier cosine series of \(f(x)=x-\frac{1}{2}\) Problem
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
Fundamental Theorem of Calculus, The Theorem
GLBP Problem
Greatest Lower Bound Property Problem
Greatest Lower Bound Property (GLBP)
definition of Problem
Gregory, James
portrait of Figure
Gödel, Kurt
portrait of Figure
Halmos, Paul Paragraph
portrait of Figure
fundamental solutions of Problem
parameter \(k\) must be less than zero Problem
solving for \(\xi(x)\) Problem
Hilbert, David
portrait of Figure
image
definition Definition (see also pre–image)
infinite sets
uncountable Paragraph
\(\infty\)
divergence to Definition
negative infinity
divergence to Definition
positive infinity
divergence to Definition
a polynomial with odd degree must have a root Problem
continuity and Paragraph
the case \(f(a)\leq v\leq f(b)\) Problem
portrait of Figure
Lagrange’s form of the remainder Theorem
\(\ln 2\) Problem
\(x\lt a\) Problem
Least Upper Bound Definition
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 Problem
Lebesgue, Henri Paragraph
portrait of Figure
Leibniz, Gottfried Wilhelm Paragraph Paragraph Paragraph Paragraph Quotation Paragraph Paragraph Paragraph
and infinitesimals Paragraph
differentiation rules Paragraph
first Calculus publication Paragraph
portrait of Figure
limit Definition
\(\limitt{x}{0}{\frac{\sin
x}{x}}=1\) Problem
\(\limit{x}{a}{\frac{x^2-a^2}{x-a}}=2a\) Problem
accumulation point Problem
accumulation points Definition
and sequences Problem
definition of non–existence Problem
identifing the theorems used in a limit Problem
of a constant sequence Problem
of a constant times a sequence Problem
of a product of sequences Theorem
of a quotient of sequences Theorem
of a sum of sequences Theorem
of interval endpoints in the NIP Theorem
of ratios of polynomials Problem
of the difference of sequences Problem
products of Theorem
Squeeze Theorem Problem
Squeeze Theorem for Sequences Theorem
verify limit laws from Calculus Problem
verifying limits via continuity Problem
linear order
definition Definition
LUB. See Least Upper Bound
Maclaurin, Colin
portrait of Figure
multiplication of Cauchy sequences Definition
multiplication of Dedekind cuts Definition
multiplication of positive Dedekind cuts Definition
near Definition
Nested Interval Property
implies the existence of square roots of integers Problem
implies the LUBP Problem
square roots of integers, and Problem
foundation of Calculus Paragraph
portrait of Figure
NIP. See Nested Interval Property
number field
\(\CC\) is a field Problem
\(\QQ\) is a field Problem
for Cauchy sequences Problem
linearly order Definition
linearly ordered Problem
open set
definition of Definition
ordering of Dedekind cuts Definition
orthogonality
of \(\cos nx\) Problem
of \(\sin nx\) Problem
partition refinement Definition
\(\pi\)
first series expansion of Problem
second series expansion Problem
Pointwise Convergence Definition
pointwise convergence Problem
derivative and Theorem
Polya, George
portrait of Figure
polynomials
infinite Paragraph
with odd degree must have a root Problem
Portraits
Abel Figure
Bolzano Figure
Cantor Figure
Cardano Figure
Cauchy Figure
Cohen Figure
Darboux Figure
Dedekind Figure
Dirichlet Figure
Euler Figure
Fourier Figure
Gregory Figure
Gödel Figure
Halmos Figure
Hilbert Figure
Johann Bernoulli Figure
Lagrange Figure
Lebesgue Figure
Leibniz Figure
Maclaurin Figure
Newton Figure
Polya Figure
Riemann Figure
Russell Figure
Tartaglia Figure
Taylor Figure
Vitali Figure
Weierstrass Figure
Wiles Figure
power series
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 Definition
uniform convergence of Problem
Weierstrass–\(M\) Test and Problem
pre–image
definition Definition (see also image)
Problems
\(f(x) = mx +b\) is continuous everywhere Problem
a closed interval is a closed set Problem
a closed set is not (necessarily) a closed inteval Problem
Abel’s Lemma Problem
Abel’s Partial Summation Formula Problem
Abel’s Theorem Problem
absolute convergence implies convergence Problem
absolute convergence of rearrangements Problem
absolute convergence of vs. the absolute value of a series Problem
absolute value Problem
accumulation point Problem
all subsequences of a convergent sequence converge Problem
antiderivative
Does every continuous function have an antiderivative? Problem
Archimedean Property and \(\QQ\) Problem
is a Taylor series Problem
Bolzano–Weierstrass Theorem
implies that a continuous functions on a closed set is bounded Problem
implies the NIP Problem
proof of Problem
bounded, non–decreasing sequences Problem
Cantor set is uncountable Problem
Cantor’s middle–third set has measure zero Problem
Cantor’s Theorem Problem
Cauchy Criterion, the Problem
Cauchy Form of the Remainder Problem
don’t always converge in \(\QQ\) Problem
Cauchy’s counterexample
part 1 Problem
part 2 Problem
Cauchy’s flawed proof that the limit of continuous functions is continuous Problem
common denominators Problem
commuting integral an limit Problem
\(\CC{}\)
is a number field Problem
constant functions are continuous Problem
implies integrability Problem
via sequences Problem
via set theory Problem
continuity in set theory Problem
continuous functions Problem
continuous non–differentiable function Problem
convergence of Cauchy sequences Problem
convergence of Cauchy sequences is equivalent to the NIP Problem
convergence of the absolute value of a sequence Problem
convergence/divergence of the Geometric series Problem
convergent sequences are bounded Problem
convergent sequences are Cauchy Problem
countable sets Problem
unions and intersections of Problem
countable union of countable sets is countable Problem
countable union of finite sets Problem
countably infinite subsets Problem
creating irrationals from rationals Problem
defining infinite decimal addition Problem
definition of continuitycon Problem
definition of divergence Problem
definition of non–existence of a limit Problem
deleting a countable subset from an uncountable set Problem
Derivative
\(f^\prime =0 \imp f\) is constant Problem
differentiability implies continuity Problem
differentiation
of \(\sin x\) as a power series Problem
differentiation of a sequence of functions Problem
differentiation rules Problem
Dirichlet function is simple Problem
Dirichlet’s Function Problem
divergence to \(\infty\) Problem
divergent Fourier series example Problem
Euler’s Formula Problem
every Cauchy sequence is bounded Problem
expand \(e^{-x^3}\) as a power series Problem
\(e^x\) is continuous everywhere Problem
Extreme Value Theorem Problem
Fermat’s Theorem Problem
find a bounded sequence of rational numbers such that no subsequence converges to a rational number Problem
Fourier Series
computing the coefficients of Problem
term by term differentiation of Problem
Fourier Series and uniform convergence Problem
Fundamental Theorem of Calculus Problem
Geometric Sequence Problem
Geometric series Problem
Geometric series as a Taylor series Problem
Greatest Lower Bound Property Problem
Heat Equation, the Problem
fundamental solutions of Problem
solving for \(\xi(x)\) Problem
Heaviside’s function is not continuous at zero Problem
if the sequence \(\left(s_n\right)\) is bounded then \(\limit{n}{\infty}{\left(\frac{s_n}{n}\right)}=0\) Problem
integration and uniform convergence Problem
Intermediate Value Theorem (IVT) Problem
Intermediate Value Theorem, the Problem
intervals are uncountable Problem
irrational numbers Problem
Lagrange’s idea Problem
Least Upper Bound Property Problem
implies the existence of irrational numbers Problem
Least Upper Bound Property doesn’t hold in \(\QQ\) Problem
Least Upper Bound Property implies the Archimedean Property Problem
Least Upper Bound Property implies the Nested Interval Property Problem
Lebesgue’s Dominated Convergence Theorem implies Arzelà’s Bounded Convergence Theorem Problem
\(\limitt{x}{0}{\frac{\sin
x}{x}}=1\) Problem
\(\limit{x}{a}{\frac{x^2-a^2}{x-a}}=2a\) Problem
limit
properties of Problem
limit is unique Problem
limit of a constant sequence Problem
\(\ln x\) is continuous everywhere Problem
lower and upper bounds for sequences Problem
Mean Value Theorem, the Problem
measure zero Problem
implies the LUBP Problem
square roots of integers, and Problem
weak form Problem
\(n\)th term test Problem
number fields
\(\QQ{}\) Problem
\(\CC{}\) Problem
linearly ordered Problem
properties of Problem
open covers Problem
open intervals are not the only open sets Problem
open sets Problem
orthogonality of \(\cos (nx)\) Problem
orthogonality of \(\sin x\) Problem
outer measure Problem
partition of \([a,b]\) Problem
partition refinement Problem
pointwise convergence Problem
power rule with fractional exponents Problem
power series expanded about \(a\) Problem
power series expansion of Problem
power series expansion of \(\arctan x\) Problem
power series expansion of \(\frac1x \) Problem
power series expansion of \(\pi \) Problem
power series expansion of \(\sqrt{1+x}\) Problem
power series expansion of \(\sqrt{x}\) Problem
power series expansion of \(a^x\) Problem
power series expansion of \(x^3+2x^2+3\) Problem
power set Problem
products of rationals and irrationals Problem
Prove that \(\left(n\right)_{n=1}^\infty \) diverges Problem
Prove that \(\underline{\int^b_{x=a}}{f(x)}\dx{x}\le
\overline{\int^{b}_{x=a}}{f(x)\dx{x}}\) Problem
Prove that \(e^{a+b}=e^ae^b\) Problem
Prove that divergence to infinity implies divergence. Problem
Prove that the limit of a constant function is the constant. Problem
Prove that uniform continuity implies continuity Problem
Prove the Fundamental Theorem of Calculus Problem
\(\QQ\) is countable Problem
Ratio Test Problem
rational numbers exist between rational numbers Problem
\(\RR\) defined by Cauchy sequences Problem
rearrangements of the Alternating Harmonic Series Problem
Riemann integrability implies Lebesgue integrability Problem
Riemann integral with countably infinitely many discontinuities Problem
Riemann integral with finitely many discontinuities Problem
Rolle’s Theorem Problem
series
solutions of \(\frac{\dx^2y}{\dx{ x}^2}=-y\) Problem
sets that are neither open nor closed Problem
simple functions Problem
\(\sin e^x\) is continuous everywhere Problem
sine series of an odd function Problem
\(\sqrt{x}\)
is continuous at every positive real number Problem
\(\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{...}}}}\)
value of Problem
\(\sqrt{x}\) is continuous at zero Problem
Squeeze Theorem for functions Problem
Strong Cauchy criterion, the Problem
sums and products of rational and irrational numbers Problem
Taylor series Problem
Taylor’s series used to approximate \(\ln 2\) Problem
Taylor’s Theorem Problem
term by term derivative of power series Problem
term by term differentiation of power series Problem
termwise products of sequences Problem
termwise quotient of sequence Problem
term–by–term integration of power series Problem
the Basel Problem Problem
the Brachistochrone problem Problem
the Comparison Test Problem
the composition of continuous functions is continuous Problem
the difference of sequences Problem
the Fourier cosine series of \(f(x)=x-\frac{1}{2}\) Problem
the limit of a constant times a sequence Problem
the limit of ratios of polynomials Problem
the product of continuous functions is continuous Problem
the quotient of continuous functions is continuous Problem
the sequence of positive integers diverges to infinity Problem
The Squeeze Theorem Problem
the Topologist’s sine function is continuous at zero Problem
The Triangle Inequality Problem
Topologist’s sine function
modified version is not continuous at zero Problem
Triangle Inequalities
for Integrals Problem
uncountably many discontinuities Problem
uniform continuity Problem
uniform convergence and continuous functions Problem
uniform convergence and integration Problem
uniform convergence and positive power series Problem
uniform convergence of power series Problem
verifying limits Problem
Weierstrass–\(M\) Test and power series Problem
Quadratic Formula
first proof Problem
second proof Problem
\(\QQ\)
creating irrationals from rationals Problem
creating rationals from irrationals Problem
rational numbers exist between rational numbers Problem
sums and products of rational and irrational numbers Problem
>\(\RR\)
the number \(1\) as a Cauchy sequence Problem
\(\RR\)
addition of Cauchy sequences Problem
any complete, linearly ordered field is isomorphic to Theorem
as Cauchy sequences
identify the multiplicative identity Problem
defined by Cauchy sequences Problem
defining infinite decimal addition Problem
irrational numbers Problem
irrational numbers drill, 5 parts Problem
is uncountable
Cantor’s first proof Theorem
ordering Dedekind cuts Problem
products of rationals and irrationals Problem
real numbers exist between real numbers Theorem
Real numbers
as Cauchy sequences Definition
Remainder of the Taylor Series
Riemann Integral Definition
Riemann, Bernhard Paragraph
portrait of Figure
Rolle’s Theorem Theorem
Russell, Bertrand
portrait of Figure
Russell’s Paradox Russell’s Paradox
Sequence vs. Series Section
sequences
all subsequences of a convergent sequence converge Problem
bounded and non–decreasing Problem
Cauchy sequences Definition
addition and multiplication of Definition
addition of is well defined Problem
Cauchy’s remainder Theorem
don’t always converge in \(\QQ\) Problem
equivalent Problem
every Cauchy sequence is bounded Problem
field axioms for Problem
real numbers as Cauchy sequences Definition
zero as a Cauchy sequence Theorem
constant multiples of Problem
constant sequences Problem
convergence Definition Theorem Problem
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
equivalent Cauchy sequences Definition
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 \(\limit{n}{\infty}{\left(\frac{s_n}{n}\right)}=0\) 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
the sequence of positive integers diverges to infinity Problem
series
\(n\)th term test Problem
absolute convergence of
vs. the absolute value of a series Problem
Alternating Harmonic Series
Binomial series
is a Taylor series Problem
Binomial series, the Theorem
Cauchy Criterion
Strong Cauchy criterion Problem
Cauchy Criterion, the Problem
Cauchy sequences
Cauchy Criterion Theorem
Comparison Test Theorem
Geometric Sequence
divergence condition Problem
Geometric series Problem
alternating Problem
as a Taylor series Problem
convergence/divergence conditions for Problem
derivation of the series representation of \(\ln(1+x)\) from Problem
differentiating Problem
naive derivation Paragraph
used to derive arctangent series Problem
graph the square root power series Problem
slow divergence of Paragraph
of \(\arctan x\) Problem
rearrangements Theorem
solutions of \(\frac{\dx^2y}{\dx{ x}^2}=-y\) Problem
Taylor’s series Theorem
\(f^{(n)}\lt B,\forall\ n\in\NN\imp\) Taylor series converges Problem
Cauchy Form of the Remainder Problem
drill problems Problem
used to approximate \(\ln 2\) Problem
the Comparison Test Problem
sets
accumulation points Definition Problem
cardinality of a power set Problem
countably infinite subsets Problem
derived sets Definition
intervals are uncountable Problem
power set Problem
\(\sin x\)
derivative of the power series representation Problem
is continuous for \(0\leq x\lt \frac{\pi}{2}\) Problem
orthogonality of Problem
Taylor’s series for Problem
\(\sqrt{2}\)
is irrational Paragraph
meaning of Paragraph
\(\sqrt{x}\)
is continuous at every positive real number Problem
is continuous at zero Problem
square roots exist Theorem
Squeeze Theorem
subsequences Definition
subtraction of Dedekind cuts Definition
Tartaglia
portrait of Figure
Taylor, Brook
portrait of Figure
drill problems Problem
use to obtain the general binomial series Problem
The Archimedean Property Theorem
Topologist’s sine function
is continuous at zero Problem
modified version is not continuous at zero Problem
Triangle Inequalities Lemma
for Integrals Problem
Reverse Triangle Inequalitiy Lemma
uncountable sets
deleting a countable subset Problem
Uniform Continuity Definition
Uniform Convergence Definition
uniform convergence Definition
Fourier Series and Problem
of power series at the endpoints of the interval of convergence Problem
positive power series and Problem
power series and Problem
Upper Bound Definition
Vitali, Giuseppe
portrait of Figure
