Real Analysis
Sets IntroPermalink
Def: A set is a collection of objects known as elements
- the set with no element is known as the empty set, ∅={}
- two sets are euqal if they contain exactly the same elements, A=B if x∈A <=> x∈B
- the cardinality of a set is the number of elements it contains, write |A|
Common Sets of Numbers
- ℕ = {1,2,3,…}, |ℕ|=∞
- ℤ = {…,-2,-1,0,1,2,…}, |ℤ|=∞
- ℚ = {p/q | p,q∈ℤ, q≠0}
- ℝ = all real numbers, the precise definition is tricky, |ℝ|>|ℕ|
-
ℂ = {a+bi | a,b∈ℝ, i^2=-1}, complex number
ex. A={orange, blue}, |A|=2 elements are colors
B={ℕ,ℝ}, |B|=2 elements are infinitely sets
Describing Sets
- Roster Strategy, {1,2,3} <- list all the elements
- Verbally, ‘All natural numbers less than 4’
-
Set Builder
{general shape of an element | specific rule it satisfies} s.t./column
examples:
ex. {n∈ℕ | n<4}
even integers = {2n | n∈ℤ} = {m | m=2n, n∈ℤ}
odd integers = {2n+1 | n∈ℤ} = {m | m=2n+1, n∈ℤ}
primes = {p∈ℕ | p is prime} = {p∈ℕ | if p=ab then a or b=1}
prefect cubes = {n^3 | n∈ℤ} = {m | m=n^3, n∈ℤ}
solutions to x^2 = {x∈ℝ | x^2=2} = {√2, -√2}
if {x∈ℚ | x^2=2} = ∅
intervals:
(3,6] = {x∈ℝ | 3<x≤6}
[a,∞) = {x∈ℝ | x≥a}
{m∈ℤ | |m|≤3} = {m∈ℤ | -3≤m≤3} = {-3,-2,-1,-,1,2,3}
{m∈ℝ | |m|≤3} = [-3,3]
Q: what is A={3a+2b | a,b∈ℤ}?
Cartesian ProductsPermalink
The Supremum and Infimum of ℝPermalink
Def: a set A⊆ℝ is bounded above, if ∃u∈ℝ s.t. a≤u ∀a∈A, u is called an upper bound
Def: a set A⊆ℝ is bounded below, if ∃l∈ℝ s.t. a≥l ∀a∈A, l is called a lower bound
Def: s∈ℝ is the least upper bound (supremum) of A⊆ℝ, if
- s is an upper bound i.e. ∀a∈A, s≥a
- if u is also an upper bound, then u≥s i.e. if ∀a∈A, u≥a then u≥s
Def: i∈ℝ is the greatest lower bound (infimum) of A⊆ℝ, if
- i is a lower bound i.e. ∀a∈A, i≤a
- if l is also a lower bound, then i≥l i.e. if ∀a∈A, l≤a then i≥l
Lemma: assume s∈ℝ is an upper bound of A⊆ℝ, then s=sup(A) iff
for all ε>0, there is a∈A s.t. s-ε<a
Axiom of CompletenessPermalink
Every nonempty set A⊆ℝ w/ an upper/lower bound has a least upper bound/greatest lower bound
This is not ture in ℚ:
ex. A = {x∈ℚ | x^2=2}
A is bounded above by 2,3,4
sup(A) = √2 ∉ ℚ
so the numbers in ℚ are not complete
B = {a_n | n∈ℕ}
a_0=3, a_1=3.1, a_2=3.14, a_3=3.141, ...
sup(A) = π ∉ ℚ
Nested Interval PropertyPermalink
Theorem: for each n∈ℕ, sps we have a closed interval In=[an,bn]={x∈ℝ | an≤x≤bn}
s.t. …⊆I3⊆I2⊆I1 (i.e. I{n+1}⊆In, n-lower label)
then ∩In ≠ ∅ (sub label of intersection is n∈ℕ)
Why not in ℚ?
Why not in open interval?
Archimedean PropertyPermalink
Theorem:
-
given any x∈ℝ, ∃n∈ℕ s.t. n>x
-
given any y∈ℝ w/ y>0, ∃n∈ℕ s.t. 1/n<y
Density of ℚ in ℝPermalink
Theorem: for any a,b∈ℝ w/ a<b, ∃r∈ℚ s.t. a<r<b
EquinumerosityPermalink
Def: two sets A,B are said to be equinumerous (have the same cardinality)
if there is a bijective funciton f: A->B
write: |A|=|B|
A is said to be countable if it is finite or |A|=|ℕ| (A is equinumerous with ℕ)
ex. |ℤ|=|ℕ|
ex. sps a,b∈ℝ, (a≠b), |(a,b)|=|ℝ|
step1. to show |(a,b)|=|(0,1)|
step2. to show |(-1,1)|=|ℝ|
Intermediate Value TheoremPermalink
The Countability of ℚPermalink
The Uncountability of ℝPermalink
|ℝ|=|P(ℕ)|Permalink
SequencePermalink
Def: a sequence is a function whose domain in ℕ
i.e. a: ℕ -> ℝ
write: a_n = a(n), {a_1,a_2,a_3,…}
Convergence and DivergencePermalink
Def: we say a sequence {a_n}^∞_{n=1} converges to L
if for every ε>0 there is N∈ℕ s.t. |a_n-L|<ε for n≥N
i.e. ∀ε>0, ∃N∈ℕ. s.t. |a_n-L|<ε for n≥N
write
lim a_n = L
n->∞
A sequance that does not converge is said to diverge
by negating Def of Convergence, we have Def of Divergence
Def: we say {a_n}^∞_{n=1} diverges
if for all L∈ℝ, there exists ε>0 s.t. for all N∈ℕ we have n≥N
but |a_n - L| ≥ ε
ε-N ProofPermalink
stratch work:
- manipulate |a_n - L| < ε <- goal
- until n > [some stuff w/ ε’s] <- call this N
proof:
- given ε>0
- set N = [some stuff w/ ε’s]
- observe that if n≥N
- then …
- we have |a_n - L| < ε
examples
ex. lim 1/n^2 = 0
n->∞
lim (1-1/n) = 1
n->∞
Convergent => BoundedPermalink
Theorem: Every Convergent Sequence is Bounded
However, A bounded sequence may not converge
ex. a_n = (-1)^n
Algebraic Properties of LimitsPermalink
Recall Def: {a_n}^∞_{n=1}
lim a_n = L ≠ ±∞
n->∞
if ∀ε>0, ∃N∈ℕ s.t. if n≥N, |a_n - L|<ε
Recall Theorem:
if {a_n}^∞_{n=1} converges, then it is bounded
i.e. ∃M>0, M∈ℝ, s.t. |a_n|<M ∀n∈ℕ
Triangle Inequality:
∀x,y∈ℝ, |x+y|≤|x|+|y|
Theorem: sps lim a_n = A, lim b_n = B, c∈ℝ except {0}, note lim under-label is {n->∞}
- lim ca_n = cA
- lim a_n+b_n = A+B
- lim a_n*b_n = AB
- lim a_n/b_n = A/B
Monotone + Bounded => ConvergentPermalink
Def: we say {a_n} is
monotonically increasing if a_{n+1}≥a_n ∀n∈ℕ
monotonically decreasing if a_{n+1}≤a_n ∀n∈ℕ
Theorem: a monotone sequence {a_n} converges iff it is bounded
ex.
SeriesPermalink
Def: let {a_n}^∞_{n=1} be a sequence, an infinite series is in the form
∞
Σ a_n
n=1
(adding up all elements of the sequence)
Def: the sequence of partial sums (seq of par sum/a companian seq)
N
S_N = Σ a_n = a_1 + a_2 + ... + a_n
n=1
We say Σ^∞_{n=1} a_n converges to S if lim_{n->∞} S_N = S
SubsequencePermalink
Def:
Given sequence
∞
{a_n} ∈ℝ
n=1
∞
{n_k} ∈ℕ usually n_1<n_2<n_3<...
k=1
the sequence
∞ ∞
{a_n_k} is called a Subsequence of {a_n} ∈ℝ
k=1 n=1
ex. a_n = 1/n, n_k = 2^k
a_n_k = 1/(2^k), k≥1
a_n: 1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, ... , 1/16, ...
a_n_1 a_n_2 a_n_3 a_n_4
∞ ∞
{a_n_k} ⊆ {a_n}
k=1 n=1
Proposition: every Subsequence of a convergent seq converges (to the same limit)
proof: sps {a_n}^∞_{n=1} s.t. lim a_n = L w/ n->∞
and {n_k}^∞_{k=1} s.t. n_1<n_2<n_3<... (strictly increasing)
to show lim a_n_k = L w/ k->∞
given ε>0
b/c {a_n} converges to L
we can find a N∈ℕ s.t. if n>N, then |a_n - L|<ε
since {n_k} is strictly increasing
there is K∈ℕ, s.t. if k≥K, n_k≥N
-> |a_n_k - L| < ε
-> lim a_n_k = L w/ k->∞ ■
Proposition: every bounded seq has a convergent subseq
proof: (assume we have a bounded seq, construct some subseq of that bounded seq that has a lim)
sps {a_n}^∞_{n=1} ∈ℝ s.t. |a_n|<M ∀n∈ℕ
-> a_n ∈ [-M,M]=[-M,0]∪[0,M] ∀n∈ℕ
so there are ∞-many a_n in [-M,0] or [0,M]
pick the second half and call it I1
pick an element in {a_n}∩I1 call it a_n_1
split I1 into 2 equal parts in mid
call the second half of ∞-many form I2
pick an element from it and call it a_n_2
ex. a_n_1 ∈ I1=[0,M]=[0,M/2]∪[M/2,M]
a_n_2 ∈ I2=[M/2,M]=[M/2,3M/4]∪[3M/4,M]
a_n_3 ∈ I3=[3M/4,M]
...
construct ...⊆ Ik ⊆... ⊆ I3 ⊆ I2 ⊆ I1
a_n_k a_n_3 a_n_2 a_n_1
∞
by Nested Interval Property ∩ Ik ≠ ∅
k=1
length(I1)=M
length(I2)=M/2
...
length(Ik)=M/2^(k-1)
∞
take L ∈ ∩ Ik
k=1
claim: lim a_n_k = L
k->∞
proof: given ε>0, take K≥0 s.t. 1/2^(K-1) < ε/M
-> M/2^(K-1) < ε
if k≥K, we have a_n_k ∈ Ik ⊆ IK w/ length(Ik)<length(IK)<ε
note both a_n_k and L ∈ Ik in length(Ik)=M/2^(k-1)
-> |a_n_k - L| < ε ■
Cauchy SequencePermalink
Def: a seq is called Cauchy seq if ∀ε>0, ∃N∈ℕ s.t. if m,n≥N then |a_m - a_n| < ε
(i.e. the very later terms of the seq are very close to each other)
ex. a_n = n/(n+1) is cauchy
A Convergent Seq <=> A Cauchy SeqPermalink
Theorem: sps {a_n}^∞_{n=1} ∈ℝ, a_n converges iff a_n is Cauchy
Cauchy Criterion for SeriesPermalink
Recall: we say an infinite series
∞
Σ a_n = A
n=1
converges (to A) if lim S_m = A w/ m->∞ where
m
S_m = a_1+a_2+...+a_m = Σ a_n
n=1
Theorem: if Σ a_n=A, Σ b_n=B, and c∈ℝ, then
- Σ c*a_n = cA
- Σ a_n+b_n = A+B
note Σ^∞_{n=1}
Theorem: the series Σ^∞_{n=1} a_n converges iff
∀ε>0, ∃N∈ℕ, s.t. if m,n≥N, |a_{m+1}+…+a_{m+n}|<ε
i.e. we can not only make the terms of the series as small as we want, we can also make sums of the series as small as we want as long as we go far enough out into the tail
Corollary:
∞
if Σ a_n converges
n=1
then lim a_n = 0
n->∞
Comparison TestPermalink
Theorem: sps 0≤a_n≤b_n, then
- if Σ b_n converges then Σ a_n converges
- if Σ b_n diverges then Σ b_n diverges
Absolute Convergence TestPermalink
Theorem: Σ|a_n| converges, then Σ a_n converges
Alternating Series TestPermalink
Theorem: sps a_1≥a_2≥a_3≥…a_n≥0, lim_{n->∞} a_n = 0, then
(n+1)
Σ(-1)^ *a_n converges
Cauchy Condensation TestPermalink
Theorem: sps {a_n}^∞_{n=1} is decreasing, and a_n≥0, ∀n∈ℕ, then
∞
Σ a_n converges
n=1
iff
∞ n
Σ 2^ * a_ converges
n=0 {2^n}
Rearrangement of Absolute Convergent SeriesPermalink
Def: the series Σ b_n is said to be a rearrangement of Σ a_n
if there is a bijection f: ℕ->ℕ, s.t. b_{f(n)}=a_n
Theorem: if Σ a_n absolutely converges
then any rearrangement Σ b_n converges to the same value
(Σ a_n absolutely converges means Σ|a_n| converges)
Open Subset of ℝPermalink
Def: given any a∈ℝ and any ε>0, the ε-neighbourhood of a is
V_ε(a) = {x∈ℝ | |x-a|<ε} = (a-ε, a+ε)
Def: a set A⊆ℝ is said to be open if ∀a∈A, ∃ε>0 s.t. V_ε(a)⊆A
ex. ℝ is open
∅ is open
all open intervals are open
(a,b) is open, w/ a,b∈ℝ
Theorem: the arbitary union of open sets is an open set
Theorem: the finite intersection of open sets is an open set
Def: a set A⊆ℝ is said to be not open if ∃a∈A s.t. ∀ε>0 we have V_ε(a)⊄A
ex. singleton sets are not open
{0} is not open
Limit PointPermalink
Def: we say x is a limit point of A⊆ℝ if ∀ε>0
V_ε(a)∩A contains at least one point other than x
i.e. [V_ε(a)\{a}] ∩ A ≠ ∅, lp=limit point
ex. [1,3]
1, 2, 3 are lps of [1,3]
∞
0 is a lp of {1,1/2,1/3,1/4,...}={1/n} = A note 0∉A
n=1
Theorem: x is a lp of A iff there is a seq {a_n}^∞_{n=1} ⊆ A s.t.
lim a_n = x, and a_n ≠ x ∀n∈ℕ
n->∞
Isolated PointPermalink
Def: an isolated point a∈A is a non-limit point of A
by negating Def of LimPoint
We say a∈A is an isolated point if ∃ε>0, s.t. V_ε(a) ∩ A = {a}
ex. 1/n is a iso-p of {1,1/2,1/3,1/4,...}={1/n}^∞_{n=1} = A
V_ε(1/n) ∩ A = {1/n}
Closed SetPermalink
Def: A⊆ℝ is said to be closed if it contains all of its limit points
ex. all closed intervals are closed
[a,b] is closed
Theorem: sps A⊆ℝ
- A is open <=> A^c is closed
- A is closed <=> A^c is open
Closure of a SetPermalink
Def: sps A⊆ℝ and L={x∈ℝ | x is a limit point of A}
then the closure of A is A-bar
_
A = A ∪ L
Theorem: A-bar is the smallest closed set containing A
Compact SetPermalink
Def: we say K⊆ℝ is compact, if for all sequence {a_n}^∞_{n=1} ⊆ K
there is a subsequence {a_n_k}^∞_{k=1}, s.t. a_n_k converges and it’s limit ∈ K
lim a_n_k = L ∈ K
k->∞
Compact <=> Closed and BoundedPermalink
Theorem: K⊆ℝ is compact iff it is closed and bounded
Nested Compact SetPermalink
Recall Nested Interval Theorem: if I_n=[a_n,b_n], a_n,b_n∈ℝ, w/ …⊆I_3⊆I_2⊆I_1, then
∞
∩ I_n ≠ ∅
n=1
Theorem: sps …⊆K_3⊆K_2⊆K_1 are nonemtpy compact sets then
∞
∩ K_n ≠ ∅
n=1
Sup and Inf of Compact SetPermalink
Recall Def: we say K⊆ℝ is compact if every sequence in K has a convergent subsequence to an element of K
Recall: K⊆ℝ is compact iff it’s closed and bounded
Proposition: sps ∅≠K⊆ℝ is compact, then sup(K) and inf(K) ∈ K
Open CoverPermalink
Def: an open cover of A⊆ℝ is a collection of open sets {U_i | i∈I} s.t.
A ⊆ ∪ U_i
i∈I
i.e. the open sets {U_i | i∈I} are open cover of A
∞
ex. an open cover of (0,4) is {(1/k,4-1/k)}
k=1
={(1,3),(1/2,4-1/2),(1/3,4-1/3),...}
another open cover of (0,4) can be
{(-∞,2),(1,3),(1,2),(0.5,3.5),(3,7)}
an open cover of [0,4] is
{(1/k,4-1/k)}^∞_{k=1} ∪ {(-0.2,0.2)} ∪ {(3.9,4.1)}
Def: if {U_i | i∈I} has a finite subset {U_i | i∈F} meaning F⊆I
which is still a cover of A, {U_i | i∈F} is called a finite subcover of A
The Heine-Borel TheoremPermalink
Theorem: a set K⊆ℝ is compact iff every open cover of K has a finite subcover
NotationPermalink
ℕ,ℤ,ℚ,ℝ,ℂ
∈ℕ,∈ℤ,∈ℚ,∈ℝ,∈ℂ
ε>0
∈ ∉ ∀ ∃ ≤ ≥ ≠ ∅ Σ Π π ∞ ⊆ ⊄ ∤ ± ≅ ≡ √ ∩ ∪ ∘ ≜
i.e. in other words
e.g. for example
ex. example
sps suppose
■ end of proof
!contradiction!
-> <- implies instead of => <=
{} under label
gcd() greatest common divisor ex. gcd(5,10)=5 gcd(24,36)=12
lcm() least common multiple ex. lcm(36,12)=36 lcm(18,12)=36
w/ with
w/o without
b/c because
equv equivalent to
\ except
A^c complement of A in U, A^c=U\A
MemoPermalink
Composition of Functions
applying one function to another
g ∘ f ≜ g(f(x))
ex. f(x)=2x+3 g(x)=x^2
g∘f(x) = (2x+3)^2
f∘g(x) = 2x^2+3
Injective: is a function f that maps distinct elements to distinct elements that is, f(x1) = f(x2) implies x1 = x2
In other words, every element of the function’s codomain is the image of at most one element of its domain
Surjective: is a function f that maps an element x to every element y that is, for every y, there is an x such that f(x) = y
In other words, every element of the function’s codomain is the image of at least one element of its domain
It is not required that x be unique; the function f may map one or more elements of X to the same element of Y
Bijective: is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set
There are no unpaired elements
In mathematical terms, a bijective function f: X → Y is a one-to-one (injective) and onto (surjective) mapping of a set X to a set Y
The term one-to-one function must not be confused with one-to-one correspondence that refers to bijective functions, which are functions such that each element in the codomain is an image of exactly one element in the domain
The term one-to-one correspondence must not be confused with one-to-one function (an injective function; see figures)
ReferencePermalink
Micheal Penn’s Youtube Channel