math 104 terms, concepts, theorems

rational numbers (Q)

field of numbers where p/q for p,q ∈ ℤ, q ≠ 0

rational zeros theorem

supposed c₀, c₁,…, c_n are integers and r is rational number (root) for c_nxⁿ + c_n-1x^n-1 + …+ c₁x + c₀ = 0 and r = c/d, c divides c₀ and d divides c_n

corollary 2.3 (solutions of polynomials)

for xⁿ + c_n-1x^n-1 + …+ c₁x + c₀ = 0, any rational solutuon of this must be an integer that divides c₀ (r = c/1)

field properties

algebraic operations that hold true for fields
ex: ab = ba, a+(b+c) = (a+b)+c, a(b+c) = ab + ac

ordered field properties

ordered properties (<, >, ≤, ≥) that are applied to ordered fields
ex: a≤b, b≤a, —> a=b; a≤b and 0≤c, —> ac ≤ bc

distance

the length between two points in space
dist(a, b) = |a - b|

triangle inequality

|a+b| ≤ |a| + |b|, ∀ a,b ∈ ℝ

maximum and minimum

M in set S such that s ≤ M for all s in set S
m in set S such that m ≤ s for all s in set S

upper and lower bounds

some M such that s ≤ M for all s in set S
some m such that m ≤ s for all s in set S
(don’t have to be in the set)

bounded sets

if there exists and upper and lower bound (trapping all numbers in set between two points)

supremum and infimum

least upper bound: least possible M such that s ≤ M for all s in set S
greatest lower bound: greatest possible m such that m ≤ s for all s in set S

completeness axiom

if a set has an upper bound, then it has a least upper bound (sup)
(not possible for Q due to the gaps in the field)

archimedean property

for a > 0 and b > 0, then there exists some n in Z+ such that na > b
(if you use a small spoon to fill bathtub, you will eventually fill the tub)

denseness of Q

for a < b and a, b in R, there exists some q in Q such that a < q < b

extended ordering

+∞ and -∞ are adjoined to R such that -∞ ≤ a ≤ +∞ for all a in set

suprema and infima for unbounded sets

+∞ and -∞ respectively

sequence

a function whose domain is a set of integers of the form {n in Z: n ≥ m}
(an infinite ordered list of numbers where every position corresponds to an integer)

convergent sequence

a sequence (s_n) where for each ε>0, there exists N such that n > N implies |s_n-s| < ε for some real number s
(terms eventually get as close) as you want to a specific target number)

divergent sequence

a sequences that does not go towards a real number
(a list of numbers that either oscillates or grows without bound and not one set value)

limit of square roots

if (s_n) is a sequence of nonnegative real numbers and s_n —> s, then √s_n —> √s

squeeze lemma

if a_n ≤ s_n ≤ b_n and lim a_n = lim b_n = s, then lim s_n = s
(if one sequence is trapped between two others that are both going to the same place, then it has no choice but to do the same and go there)

boundedness of convergent sequences

every convergent sequence is bounded

algebraic limit theorems

algebraic manipulations for limits
If s_n —> s and t_n —> t, then lim s_n + t_n —> s+t
(s_n)(t_n) —> st
s_n / t_n —> s/t

limits diverging to infinity

for sequence (s_n), we write lim s_n = +∞ provided that for each M > 0, there exists N such that n > N implies s_n > M

theorem 9.9 (infinite limits)

let (s_n) and (t_n) be sequences such that lim s_n = +∞ and lim t_{n = 0}, then lim s_nt_n = +∞

theorem 9.10 (limits)

lim s_n = +∞ if and only if lim (1/s_n) = 0

monotone sequence

a sequence that is increasing or decreasing (s_n-1 ≤ s_n) or (s_n-1 ≥ s_n)

bounded monotone convergence theorem

all bounded monotone sequences converge

lim superior and lim inferior

limit of the suprema and infima of the “tail” of the sequence as starting point N moves to infinity
lim sup s_n = lim_N—>∞ sup {s_n: n > N}
lim inf s_n = lim_N—>∞ inf {s_n: n > N}

cauchy sequence

a sequence where for each ε > 0, there exists N such that m, n > N implies |s_n - s| < ε

convergence and cauchy sequences

sequences are convergent if and only if they are cauchy

subsequences

a sequence of the form (t_k)_(k∈N₎ where for each k there is a positive integer n_k such that n₁<n₂<…<n_k<n_k+1… and t_k=s_{(n_k)}
(new list of numbers made by picking terms from original list in original order, but skipping as many as you like)

subsequential limits (theorem)

a real number t of (s_n) if and only if the set {n∈N: |s_n-t| < ε} is infinite for every ε > 0

limit of subsequences (theorem)

if sequence (s_n) converges, then every subsequence converges to the same limit

existence of monotonic subsequences

every sequence has a monotonic subsequence

Bolzano-Weierstrass Theorem

every bounded subsequence (in R^k too) has a convergent subsequence

subsequential limit

any real number or symbol +∞ or -∞ that is the limit of some subsequence of (s_n)

limit superior theorem

if (s_n) converges to a positive number s, then lim sup (s_nt_n) = s*lim sup (t_n)

comparison of ratios and roots

lim inf |(s_n+1)/(s_n)| ≤ lim inf |s_n|^1/n ≤ lim sup |s_n|^1/n ≤ lim sup |(s_n+1)/(s_n)|
(if lim |(s_n+1)/(s_n)| exists, then lim |s_n|^1/n exists)

metric space

a set S together with distance function d that satisfies:
1. d(x, y) = 0 iff x = y
2. d(x, y) = d(y, x)
3. d(x, z) ≤ d(x, y) + d(y, z)

complete metric space

a metric space is complete if every Cauchy sequence converges to a point within the space

Euclidean k-space R^k is

complete

interior point

s₀ ∈ E if for some r >0 we have E⁰ = {s ∈ S: d(s, s₀) < r} ⊆ E for some metric space, E ⊆ S

open set

a set where every point in the set is an interior point
(the interior point has a tiny ball of radius r >0 around it that is entirely inside the set)

closed set

a set where its complement is open
(E is ____ if E = S\U where U is an open set)

Heine-Borel Theorem

A subset of R^k is compact (fully contained in a finite boundary) if and only if it is closed and bounded

infinite series

∑^∞ a_n that converges if the sequence of its partial sums converges to a real number

partial sum

the finite sum of the first n terms of a series

geometric series

∑^∞arⁿ where its partial sum is ∑^∞ar^k = a((1-r)ⁿ⁺¹)/(1-r)

a series ∑a_n satisfies the Cauchy criterion if

its sequence (s_n) of partial sums is a Cauchy Sequence

A series convreges if and only if

it satisfies the Cauchy criterion

comparison test

if 0 ≤ |b_n| ≤ a_n and ∑a_n converges, then ∑b_n converges (BC, SD)

ratio test

a series ∑a_n of nonzero terms:
converges absolutely if lim sup |(a_n+1)/a_n| < 1
diverges if lim inf |(a_n+1)/a_n| > 1
lim inf |(a_n+1)/a_n| ≤ 1 ≤ lim sup |(a_n+1)/a_n|, and test gives no info

root test

let ∑a_n be a series and let α = lim sup |a_n|^1/n. The series ∑a_n:
converges absolutely if α < 1
diverges if α > 1

p-series test

the series ∑(1/n)^p converges if and only if p > 1

alternating series theorem

if a_n is a decreasing sequence of positive numbers that goes to 0, then the series ∑(-1)ⁿ⁺¹a_n converges

integral test

a series ∑a_n converges if the area under the curve of its corresponding function ∫f(x)dx is finite

existence of decimal expansions

every nonnegative real number x can be represented as k.d₁d₂d₃…, shorthand for k + ∑^∞d_j(10^-j) where k is in Z and d_j is in {0, 1, …, 9}
(any number can be written as an infinite string of digits)

A real number has exactly one…

decimal expansion unless it ends in a string of all 0s (1.000… = 0.999…)

A real number is rational if and only if…

its decimal expansion is repeating

continuity at a point

f is ____ at x₀ if for every sequence (x_n) that converges to x₀, the sequence of results f(x_n) converges to f(x₀)
(if small changes in input results in predictably small changes in output with no sudden jumps)

epsilon-delta continuity

f is continuous at x₀ in dom(f) if and only if for each ε > 0, there exists δ > 0 such that x in dom(f) and |x - x₀| < δ implies |f(x) - f(x₀)| < ε

algebraic properties of continuous functions

if f (and g) is continuous at x₀ in dom(f):
|f| and kf are continuous at x₀

f+g, fg, and f/g (for g(x₀) ≠ 0) are continous at x₀

if f is continuous at x₀ and g is continuous at f(x₀)…

g∘f is continuous at x₀

extreme value theorem (EVT)

a continuous function on a closed interval [a, b] is bounded and always achieves max and min values
(there exists x₀, y₀ in [a, b] such that f(x₀) ≤ f(x) ≤ f(y₀) for all x in [a, b])

intermediate value theorem (IVT)

if f continuous on an interval and y is any value between f(a) and f(b), there must be at least one x between a and b where f(x) = y

A continuous one-to-one function on an interval must be…

strictly increasing or strictly decreasing, and inverse will also be continuous

uniform continuity

f is _____ on set S if for each ε > 0, there exists δ > 0 such that x, y in S and |x - y| < δ imply |f(x) - f(y)| < ε
(for each ε > 0, a δ > 0 can be chosen that works for all x, y in S)

if f is continuous on a closed interval [a, b], then…

f is uniformly continuous on [a, b]

if f is uniformly continuous on a set S and (s_n) is a Cauchy sequence in S, then…

(f(s_n)) is a Cauchy sequence

if f is differentiable and its derivative f’ is bounded, then…

f is uniformly continuous

limit of a function

lim_x—>a^s f(x) = L if f is a function defined on S ⊆ R, and for every sequence (x_n) in S is with limit a, and lim_n—>∞ f(x) = L
right hand limit: lim_x—>a⁺ f(x)
left hand limit: lim_x—>a^- f(x)

Let f be function for lim_x—>a^s f(x) = L exists and is finite. If g is a function on {f(x): x in S} ∪ {L} that is continuous at L, then…

lim_x—>a^s g∘f(x) exists and equals. g(L)
(if limit of inner function f(x) is L and outer function g(x) is continuous on L, then g(f(x)) is headed toward g(L))

epsilon-delta for limits

lim_x—>a^s f(x) = L if and only if for each ε > 0 there exists a δ > 0 such that x in S and |x - a| < δ imply |f(x) - L| < ε

lim_x—>a f(x) exists if and only if…

lim_x—>a⁺ f(x) and lim_x—>a^- f(x) (right and left hand limits) both exist and are equal
(consequently, all three would be equal)

continuity between metric spaces

A function f: S—>S^* is continuous at s₀ if for each ε > 0 there exists a δ > 0 such that d(s, s₀) < δ implies d^*(f(s), f(s₀)) < ε

a function between metric spaces is continuous if and only if…

the inverse image of every open set is open
(Given (S, d) and (S^*, d^*), f:S—>S^* is continuous on S iff f^-1(U) = {s in S: f(s) in U} is open subset of S for every open subset U of S^*)

for f:S—>S^* and E compact (well-behaved, limited) subset of S, then:

1) f(E) is a compact subset of S
2) f is uniformly continuous on E

denseness

a subset D of metric space S where every nonempty open set U ⊆ S intersects D (D∩U ≠ ∅)
(ex: Q is dense in R)

nowhere dense

subset E ⊆ S when its closure E^- has an empty interior (closure of set lacks “fatness”)

E is nowhere dense if and only if…

E^- is nowhere dense

empty interior

when a set does not contain any non-empty open sets (it’s thin, and has no “fatness”)
examples in R: Q, Z, Cantor set, finite sets

E^-

a set of points in closure of E
(a point s is in closure of set E if every open set containing contains an element of E)

category 1 (of subsets of S)

sets that are unions of sequences of nowhere dense subsets of S
(meager and small)

category 2 (of subsets of S)

all other subsets of S that are not in category 1 (non-meager, large, substantial sets)

baire category theorem

a complete metric space (S, d) is of the second category in itself

(complete metric space is like a solid block of wood, and a nowhere dense set is like a tiny splinter or speck of sawdust. you can’t recreate an entire sold block just by gluing together a countable list of those splinters)

disconnected set

a set that can be separated by two open sets into disjoint, nonempty pieces, and one of the two equivalent conditions hold:
1) there exists open subsets U₁ and U₂ of S such that (E∩U₁) ∩ (E∩U₂) = ∅ and E = (E∩U₁ ∪ (E∩U₂)), E∩U₁ ≠ ∅ and E∩U₂ ≠ ∅

2) there exists disjoint nonempty subsets A and B of E such that E = A∪B and neither set intersects the closure of the other set (A^-∩B = ∅, A∩B^- = ∅)

connected set

a set that is not disconnected

if E is a connected subset of S, then…

f(E) is a connected subset of S^*

path-connected sets

for every pair s, t of points in E, there exists a continuous function γ:[a, b] —> E such that γ(a) = s and γ(b) = t (γ is a path)
(a set where every pair of points in it can be joined by a continuous path)
(you can draw a line between any two points in the set without ever lifting your pen or leaving the set)

C(S)

set of all bounded continuous real-valued functions on S, and for f, g in this set, d(f, g) = sup{|f(x) - g(x)|: x in S}

power series

∑^∞_n=0 a_nxⁿ
(can be written ∑^∞_n=0 a_n(x - x₀)ⁿ for fixed x₀ in R, IOC is centered at x₀)

radius of convergence

R = 1/β, where β = lim sup |a_n|^1/n
every power series has R such that it converges when |x| < R and diverges if |x| > R

pointwise convergence

a sequence of functions f_n converges this way to function f defined on S if lim_n—>∞ f_n(x) = f(x) for all x in S
(written f_n = f or f_n —> f)

uniform convergence

f_n —> f ____ on S if for each ε > 0, there exists a single N that works for all x in S and all n > N to make |f_n(x) - f(x)| < ε
(the entire graph of the function settles down at the same time, speed of settling is consistent across domain)

the uniform limit of continuous functions is…

continuous

what is the difference between uniform continuity and uniform convergence?

uniform continuity: the property of a single function; pick one δ that works for the entire set S (ex: when hiking, it guarantees path will never get infinitely steep, and you can determine how far to step for controlled change in altitude)
uniform convergence: property of a sequence of functions; pick N (speed) that works for entire graph (ex: when painting, entire painting starts looking like the final version all at once)

if f_n —> f uniformly on [a, b], then…

limit of integrals = integral of limit
lim_n—>∞ ∫_a^b f_n(x)dx = ∫_a^b f(x)dx

uniformly cauchy

a sequence of functions get uniformly close to each other as n and m get larger
(f_n) defined on S ⊆ R, for each ε > 0 there exists number N such that |f_n(x) - f_m(x)| < ε for all x in S and all m, n > N