Analysis Final Exam

Axiom of Completeness

Every nonempty subset of R that is bounded above has a supremum.

Upper Bound/Lower Bound

Let A subset R. b∈R is an Upper Bound for A if b≥a for all a in A. l in R is a Lower Bound for A if l≤a for all a in A

Supremum

S is the least upper bound or supremum of A if

(i) S is an upper bound for A

(ii)  If b is an upper bound for A then s≤b

Infimum

L is the greatest lower bound or infimum of A if

(i) L is a lower bound for A

(ii) If m is a lower bound for A then m≤L

m is the maximum of A 

if m is an element and upper bound i.e., m is in A and m≥a for all a in A

m is the minimum of A

if m is an element and lower bound i.e., m is in A and m≤a for all a in A

A subset R is dense in R

If for all a,b ∈ R with a < b there exists x ∈ A such that a<x<b

A has the same cardinality as B written A ~ B if there exists

f: A —> B that is bijective

A is countable

if A ~ Natural Numbers

A is uncountable

if A is infinite and not countable

Convergence of a Series

Let (bn) be a sequence. An infinite series is a formal expression of the form

bn = b1 + b2 + b3 + b4 + b5 + · · ·  from n=1 to infinity.

We define the corresponding sequence of partial sums (sm) by s_m =b1 +b2 +b3 +···+b_m, and say that the series (n=1 to infinity) bn converges to B if the sequence (sm) converges to B.

In this case, we write series (from n=1 to infinity) bn = B.

Bounded

A sequence (xn) is bounded if there exists a number M > 0 such that |xn|≤M for all n∈N

Subsequence of (a_n)

Let (an) be a sequence of real numbers and let n1 < n2 < n3 < n4 < n5 < . . . be an increasing sequence of natural numbers. Then the sequence (an1,an2,an3,an4,an5,...) is called a subsequence of (a_n_, denoted by (a_n_k), where k ∈ N indexes the subsequence.

Cauchy Sequence

for every ε > 0, there exists an N ∈ N such that whenever m,n ≥ N it follows that |am − an| < ε

Triangle Inequality

Let a,b in Real Numbers. Then |a+b|≤|a|+|b|

Triangle Inequality Properties

(i) |ab| = |a||b|

(ii) |a+b| ≤ |a| + |b|

(iii) |a-b| ≤ |a-c| + |b-c|

(iv) ||a| - |b|| ≤ |a-b|

Sup Epsilon Criterion

If s an upper bound, s=sup(A) if and only if for all epsilon>0, there exists an a in A s.t. s-epsilon<a.

Nested Interval Property

For each n ∈ N, assume we are given a closed interval In =[an,bn]={x∈R : an ≤x≤bn}. Assume also that each In contains In+1. Then, the resulting nested sequence of closed intervals has a non-empty intersection.

Archimedian Principle

(i) Given any number x ∈ R, there exists an n ∈ N satisfying n > x.

(ii) Given any real number y > 0, there exists an n ∈ N satisfying 1/n < y.

Subsets of countable sets are…

countable or finite

A countable union of countable sets…

is countable

Cantor’s Theorem

Let there be a set A. Then there exists no function f: A —> P(A) which is surjective.

Uniqueness of Limits

The limit of a sequence, when it exists, must be unique

Algebraic Limit Theorem

Let lim an = a, and lim bn = b. Then,

·      (i) lim(can) = ca, for all c ∈ R

·      (ii) lim(an +bn) = a+b;

·      (iii) lim(an*bn) = ab

·      (iv) lim(an/bn) = a/b, provided b ≠ 0.

Bolzano-Weierstrass Theorem

Every bounded sequence contains a convergent subsequence.

A series (an) converges implies

(an) → 0

Comparison Test

Assume (ak) and (bk) are sequences satisfying 0≤ak ≤bk for all k∈N.

·      (i) If sum(from k=1 to infinity) bk converges, then sum (from k=1 to infinity) ak converges.

·      (ii) If sum (from k=1 to infinity) ak diverges, then sum (from k=1 to infinity) bk diverges.

Open Set

A set O ⊆ R is open if for all points a ∈ O there exists an ε-neighborhood Vε(a) ⊆ O.

Limit Point

A point x is a limit point of a set A if every ε-neighborhood Vε(x) of x intersects the set A at some point other than x.

Closed Set

A set F ⊆ R is closed if it contains its limit points.

Closure

Given a set A ⊆ R, let L be the set of all limit points of A. The closure of A is defined to be A = A ∪ L.

Compact

A set K ⊆ R is compact if every sequence in K has a subsequence that converges to a limit that is also in K.

Open Cover

Let A ⊆ R. An open cover for A is a (possibly infinite) collection of open sets {Oλ : λ ∈ Λ} whose union contains the set A; that is, A subset U λ in Λ O_λ.

Finite Subcover

A finite subcover is a finite subcollection of open sets from the original open cover whose union still manages to completely contain A.

Functional Limit

Let f : A → R, and let c be a limit point of the domain A. We say that lim x→c f(x) = L provided that, for all ε > 0, there exists a δ > 0 such that whenever 0 < |x − c| < δ (and x ∈ A) it follows that |f (x) − L| < ε.

Continuity

A function f : A → R is continuous at a point c∈A if, for all ε>0, there exists a δ>0 such that whenever |x−c|<δ (and x ∈ A) it follows that |f(x) − f(c)| < ε.

Uniform Continuity

A function f : A → R is uniformly continuous on A if for every ε > 0 there exists a δ > 0 such that for all x, y ∈ A, |x−y|<δ implies |f(x)−f(y)|<ε.

Theorems of Collection of Open Sets

(i) The union of an arbitrary collection of open sets is open. (ii) The intersection of a finite collection of open sets is open.

Theorem for Collection of Closed Sets

(i) The union of a finite collection of closed sets is closed.

(ii) The intersection of an arbitrary collection of closed sets is closed.

Limit Point IFF

A point x is a limit point of a set A if and only if x = lim(a_n) for some sequence (an) contained in A satisfies an ≠ x for all n ∈ N.

Closed IFF

A set F ⊆ R is closed if and only if every Cauchy sequence contained in F has a limit that is also an element of F.

Complement of a Set (Open/Closed)

A set O is open if and only if O^c is closed. Likewise, a set F is closed if and only if F^c is open.

Heine-Borel Theorem

Let K be a subset of R. All of the following statements are equivalent:

(i) K is compact.
(ii) K is closed and bounded
(iii) Every open cover for K has a finite subcover.

Nested Compact Set Property

K1 ⊇K2 ⊇K3 ⊇K4 ⊇··· is a nested sequence of nonempty compact sets, then the intersection (from n=1 to Kn) is not empty.

Sequential Criterion for Functional Limits

Given a function f : A → R and a limit point c of A, the following two statements are equivalent:

(i) lim x→c f(x) = L

(ii) For all sequences (xn) ⊆ A satisfying xn ≠ c and (xn) → c, it follows that f(xn) → L.

Divergence Criterion for Functional Limits

Let f be a function defined on A, and let c be a limit point of A. If there exist two sequences (xn) and (yn) in A with xn ≠ c and yn ≠ c and

lim xn = lim yn = c but limf(xn) ≠ limf(yn),

then we can conclude that the functional limit lim x→c f(x) does not exist.

Algebraic Limit Theorem for Functional Limits

Let f and g be functions defined on a domain A ⊆ R, and assume lim x→c f (x) = L and lim x→c g(x) = M for some limit point c of A. Then,

(i) lim x→c kf(x)=kL for all k∈R,

(ii) lim x→c [f(x)+g(x)]=L+M,

(iii) lim x→c [f(x)g(x)] = LM, and

(iv) lim x→c f(x)/g(x) = L/M, provided M ≠ 0.

Compositions of continuous functions…

are continuous

Preservation of Compact Sets

Let f : A → R be continuous on A. If K ⊆ A is compact, then f(K) is compact as well.

Extreme Value Theorem

If f : K → R is continuous on a compact set K ⊆ R, then f attains a maximum and minimum value. In other words, there exist x0,x1 ∈ K such that f(x0) ≤ f(x) ≤ f(x1) for all x∈K.

Sequential Criterion for Absence of Uniform Continuity

A function f : A → R fails to be uniformly continuous on A if and only if there exists a particular ε0 > 0 and two sequences (xn) and (yn) in A satisfying |xn −yn|→0 but |f(xn)−f(yn)|≥ε0.

Uniform Continuity on Compact Sets

A function that is continuous on a compact set K is uniformly continuous on K.

Intermediate Value Theorem

Let f : [a,b] → R be continuous. If L is a real number satisfying f(a) < L < f(b) or f(a) > L > f(b), then there exists a point c ∈ (a, b) where f(c) = L.

Intermediate Value Property

A function f has the intermediate value property on an interval [a,b] if for all x < y in [a,b] and all L between f(x) and f(y), it is always possible to find a point c ∈ (x,y) where f(c) = L.

Differentiability

Let g : A → R be a function defined on an interval A. Given c ∈ A, the derivative of g at c is defined by

g’( c ) = lim x→c g(x)-g(c) / x-c

Differentiability Implies Continuity

If a function f is differentiable over [a,b], it follows that f is continuous over taht same interval

Algebraic Differentiability Theorem

Let f and g be functions defined on an interval A, and assume both are differentiable at some point c ∈ A. Then,

(i) (f+g)’(c) = f’(c) + g’(c)

(ii) (kf)′(c) = kf′(c), for all k ∈ R

(iii) (fg)′(c) = f′(c)g(c) + f(c)g′(c)

(iv) (f/g)’(c) = g(c)f’(c) - f(c)g’(c) / [g(c)]² provided that g(c) ≠ 0.C

Chain Rule

Let f : A → R and g : B → R satisfy f(A) ⊆ B so that the composition g ◦ f is defined. If f is differentiable at c∈A and if g is differentiable a tf(c)∈B,then g◦f is differentiable at c with (g ◦ f)′(c) = g′(f(c)) · f′(c).

Interior Exterior Theorem

Let f be differentiable on an open interval (a, b). If f attains a maximum value at some point c ∈ (a, b) (i.e., f(c) ≥ f(x) for all x ∈ (a,b)), then f′(c) = 0. The same is true if f(c) is a minimum value.Darboux’s Theorem

Darboux’s Theorem

If f is differentiable on an interval [a,b], and if α satisfies f′(a) < α < f′(b) (or f′(a) > α > f′(b)), then there exists a point c ∈ (a, b) where f′(c) = α.

Rolle’s Theorem

Let f : [a, b] → R be continuous on [a, b] and differentiable on (a,b). If f(a) = f(b), then there exists a point c ∈ (a,b) where f′(c) = 0.Mean Value Theorem

Mean Value Thoeorem

If f : [a, b] → R is continuous on [a, b] and differentiable on (a, b), then there exists a point c ∈ (a, b) where f’(c)= f(b)-f(a) / b-a

Generalized Mean Value Theorem

If f and g are continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists a point c ∈ (a, b) where [f(b) − f(a)]g′(c) = [g(b) − g(a)]f′(c).

Differentiability and Constant

If g : A → R is differentiable on an interval A and satisfies g′(x) = 0 for all x ∈ A, then g(x) = k for some constant k ∈ R.

Diffferentiability and Constant (Addition)

If f and g are differentiable functions on an interval A and satisfy f′(x) = g′(x) for all x ∈ A, then f(x) = g(x) + k for some constant k ∈ R.

L’Hospital’s Rule: 0/0 case

Let f and g be continuous on an interval containing a, and assume f and g are differentiable on this interval with the possible exception of the point a. If f(a) = g(a) = 0 and g′(x)≠0 for all x≠a, then

lim x→a f’(x) / g’(x) = L implies lim x→a f(x) / g(x) = L.

L’Hopital’s Rule: Infty/Infty Case

Assume f and g are differentiable on (a, b) and that g′(x) ≠ 0 for all x ∈ (a, b). If lim x→a g(x) = infty (or −infty), then

lim x→a f’(x) / g’(x) = L implies lim x→a f(x) / g(x) = L.

Uniform Convergence

Let (fn) be a sequence of functions defined on a set A ⊆ R. Then, (fn) converges uniformly on A to a limit function f defined on A if, for every ε > 0, there exists an N ∈ N such that |fn(x)−f(x)|<ε whenever n≥N and x∈A.

Cauchy Criterion for Uniform Convergence

A sequence of functions (fn) defined on a set A ⊆ R converges uniformly on A if and only if for every ε > 0 there exists an N ∈ N such that |fn(x)−fm(x)| < ε whenever m, n ≥ N and x ∈ A.

Continuous Limit Theorem

Let (fn) be a sequence of functions defined on A ⊆ R that converges uniformly on A to a function f. If each fn is continuous at c ∈ A, then f is continuous at c.

Differentiable Limit Theorem

Let f_n → f pointwise on [a,b]. Assume f_n differentiable on [a,b] and (f_n’) → g uniformly on [a,b]. Then f is differentiable and f’=g on [a,b]

Strong Form of Differentiable Limit Theorem

Let (fn) be a sequence of differentiable functions on [a,b]. Assume (fn’) → g uniformly on [a,b]. Assume there exists an x0 in [a,b] s.t. (fn(x0)) converges. Then (fn) converges uniformly to an f in [a,b] s.t. f’=g.

Series Converges Pointwise

Let fn,f be functions on A subset R for all n in N. The (sum from n=1 to infty) fn(x) converges pointwise to f(x) if for all x0 in A, the (sum from n=1 to infty) fn(x0) converges to f(x0), i.e. the sequence of partial sums lim n→infty ((sum from n=1 to k) fk(x0)) = f(x0)

Services Converges Uniformly

The (sum from n=1 to infty) fn(x) converges uniformly to f(x) if the sequence of functions Sn(x)= (sum from k=1 to N) fk(x) converges uniformly on f(x) on A.

Term by Term Continuity Theorem

fn continuous on A subset R for all n in N. Then the (sum from n=1 to infinity) fn(x) → f(x) uniformly on A, f(x) is continuous.

Cauchy Criterion for Uniform Convergence of a Series

(sum from n=1 to infty) fn(x) converges uniformly on A subset R iff for all epsilon>0 there exists N in N s.t. n>m>=N, x in A implies |(sum from k=m+1 to n) fk(x)| < epsilon.

Weierstrass M-Test

Let fn function on A subset R for all n in N and Mn is an upper bound of fn(x) on A, |fn(x)| <= Mn for all x in A. Then the (sum from n=1 to infty) Mn converges implies (sum from n=1 to infty) fn converges uniformly on A.

Partition

A partition of [a,b] is a subset P of [a,b] s.t. a,b is an element of P. P = {a=x0,x1,x2…,xn=b} x_i < x_i+1

Definition of mk(f) and Mk(f)

If f is a bounded function on [a,b] and P is a partition of [a,b], define

mk(f) = mk = inf {f(x): x in [x_k-1,x_k]}

Mk(f) = Mk = sup {f(x): xin [x_k-1, x_k]}

Lower Sum

The Lower Sum of f w/ respect to P is L(f,P) = (sum from k=1 to N) m_k(x_k-x_k-1)

Upper Sum

The upper sum of f w.r.t. P is U(f,P)= (sum from k=1 to N) M_k(x_k - x_k-1)

Refinement

A refinement of a partition P is a partition Q s.t. P subset Q.

Upper and Lower Integral

Let Fancy P= {All partitions of [a,b]}

The upper integral of f on [a,b] is U(f)= inf{U(f,P), P in Fancy P}

The lower integral of f on [a,b] is L(f) = sup(L(f,P): P in Fancy P}

Riemann integral

Let f be bounded on [a,b]. f is riemann integrable on [a,b] if U(f)=L(f). If so, the integral of f over [a,b] is

integral (from a to b) f = U(f) = L(f)

Continuity + Intergrability Theorem

f continuous on [a,b] implies f is integrable on [a,b]

Pointwise Convergence (Sequence of Functions)

For each n ∈ N, let fn be a function defined on a set A ⊆ R. The sequence (fn) of functions converges pointwise on A to a function f if, for all x ∈ A, the sequence of real numbers fn(x) converges to f(x).

Uniform COnvergence (Sequence of Functions)

Let (fn) be a sequence of functions defined on a set A ⊆ R. Then, (fn) converges uniformly on A to a limit function f defined on A if, for every ε > 0, there exists an N ∈ N such that |fn(x)−f(x)|<ε whenevern≥N andx∈A.