Math 4381

Two numbers are equals if

for all epsilon > 0, |a-b| < epsilon

The axiom of completeness

Nonempty bounded sets of real numbers have a supremum and infimum

least upper bound/greatest lower bound requirements

a is a the least upper bound (vice versa) if 1) a is an upper bound(a>x for all x in A) and 2) if b is an upper bound of A, a<b for all b

lemme 1.3.8 (epsilon def for sup and inf)

s is the supremum of A if for all e>0, there exists and x in A such that s-e<x (shaving off any piece puts you in the set). for infimum L its x>L+e

Show there is no rational number st 2^r=3

1.2.2

1.2.3 (didnt feel like writing its true false

1.2.3 answer

prove (a+b)²<= (|a|+|b|)²

1.2.6b

prove |a-b|<=|a-c|+|c-d|+|d-b|

1.2.6c

give example or explain how impossible

A set B with infB>=SupB

A finite set that contains its supremum but not infimum

A bounded subset of Q that contains its supremum but not its infimum

1.3.2

Let A in R be nonempty and bounded above, and let c in R. if c>=0, show sup(cA) = c*sup(A)

come up with a different argument for when c<0

1.3.5

Prove that if a is an upper bound for A, and a is an element of A, that it must be the supremum

1.3.7

Nested Interval property

the infinite intersection of bounded intervals is not empty

The density of Q in R (normal and sequential)

for every x,y in R, there exists a rational r such that x<r<y

sequential: for every y in the reals, there exists a sequence of rationals that converges to y

Let a<b be real numbers, consider T=Q intersected with [a,b]. show supT=b

1.4.4

one to one definition and onto definition

A function is one-to-one if every output has a distinct output. aka f(x)=f(y) iff x=y. A function is onto if for any output, there is an input. for any b in B, there exists an a in A st f(a)=b.

Two sets have the same cardinality if:

The there is a bijection (one to one and onto)

A set A is countable if

A set A is uncountable if

A~N ( A has a bijection with the naturals)

A is uncountable if it is not countable

Is Q countable or Uncountable. Same with R

Q is countable, R is uncountable

If A is a subset of B and B is countable -

A is either countable or finite

If A1…Am are countable sets, the union from 1 to m of them is…

If An is a countable set for each n in N, then the infinite union is

countable

countable

Exercise 1.5.5 (im lazy to write it) a b and c

1.5.5 answer

Give an example or explain if impossible

A non empty bounded set such that sup A < inf A

let A1,A2,A3 be infinite collection of non empty bounded ( like (a,b) ) sets such that theyre nested and the instersection is not empty

Impossible. Check Exam 1 solution

Let Ai = (0,1/n). Check exam 1 solution

Convergence of a sequence definition

A sequence xn conveges to x if for all e>0 , there exists an N in the natruals so when n>=N, |xn-x|<e

Epsilon neighborhood definition

A set of points no more than epsilon away from a point a. Ve(a) ={x in R : |x-a|<e}

thm: the limit of a sequence is

unique

topological convergence of a sequence

Xn converges to x if given any epsilon neighborhood of x, there exists a threshold N such that when n>=N, Xn is in the epsilon neighborhood

Verify the lim (2n²)/(n³+3) as n goes to infinity is 0 using the definition of convergence

2.2.2 b

Verify the lim sin(n²)/n^1/3 as n goes to infinity is =0 using definition of convergence

2.2.2 c

Prove the uniqueness of limits. Assume an→ a and an→b . argue that a=b

2.2.6

bounded definition

A sequence is bounded is there exists a M>0 such that |xn|<=M for all n in naturals (aka there is an interval [-M,M]

thm: every convergent sequence is

bounded

Algebraic limit theorem (give assumptions along with the theorem)

if an→a, bn→b, then lim c*an=c*a , lim(an+bn)=a+b

,lim(an*bn)=a*b , lim(an/bn)=a/b if b=/ 0

order limit theorem (give assumptions along with theorem)

if an→a and bn→b, then if an>=0 for all n, then a>=o

if an<=bn for all n, then a<b. and for a c in R where c<=bn for all n, then c<=b

Prove the squeeze theorem (Show that if xn<=yn<zn for all n and if limxn=limzn=L, then limyn=L too)

2.2.3

Let xn and yn be sequences and zn be shuffled sequence (x1y1×2y2…) prove that zn is convergent iff xn and yn are both convergent and limxn=limyn

2.3.5

A sequence is increasing/ decreasing if

if an <=an+1 for all n in Naturals. decreasing if an+1<=an.

Monotone definition

sequence either increasing or decreasing for all terms

monotone convergence theorem

States that every bounded monotone sequence is convergent.

convergence of a series

the series of bn converges if the sequence of partial sums converges

Cauchy condensation test

Let bn be decreasing and >=0 for all n. Then the series of bn from 1 to infinity converges if and only if the series from 0 to infinity of 2^nb2^n converges

Prove sequence defined by x1=3 and xn =1/(4-xn) converges (also do the rest of 2.4.1)

2.4.1

Show that if series 0 to infinity ogf 2^nb2^n diverges, then series 1 to infinity of bn diverges

2.4.9

subsequence definition

ank is a subsequence of an if nk<nk2<nk3 (ie the indices are strictly increasing)

subsequences of a convergent sequence -

converge to the same limit as the original sequence

divergence criterion for sequences

A sequence diverges if you can find two convergent subsequences that converge to two different limits

bolzano weirstrauss theorem

every bounded sequence contains a convergent subsequence

Assume an is a bounded sequence with the property that every convergent subsequence of an converges to the same limit a. show an converges to a too

2.5.5

cauchy sequence definition

A sequence is cauchy if for all e>0, there exists an N such that when n,m>=N, |an-am|<e

every convergent sequence is-

cauchy

cauchy criterion

a sequence is convergent iff it is cauchy

every cauchy sequence is-

bounded

prove every convergent sequence is cauchy

2.6.1 (hint: use triangle inequality)

algebraic limit theorem for series

if series ak=a and series bk=b, then series of c*ak=c*a, and series ak+bk = a+b

cauchy criterion for series

the series a_k converges iff for all e>0, there exists an N such that when n>m>=N, |a_m+1+a_m+2+..+a_{n| < e}

If the series ak converges, then

ak→0

comparison test for series

assume ak and bk are sequences where 0<=a_k<=b_k. then if the series bk converges, ak converges. if ak diverges, then bk diverges

geometric series convergence and sequence of partial sums convergence

converges to a/1-r. partial sums converges to a(1-r^m)/(1-r)

absolute convergence test

if series |a_k| converges, so does series a_k

Provide the details for the proof of the comparison test using the cauchy criterion for series

2.7.3

open definition

A set is A open if for every a in A, there exists an epsilon neighborhood for a that is contained entirely in A.

the union of an arbitrary collection of open sets is -

open

the intersection of a finite amount of open sets is

open

limit point definition (not the sequential definition)

a point is a limit point if every epsilon neighborhood for this point intersects the set A somewhere besides x

sequential definition of a limit point

a point x is a limit point iff there exists an xn=/x and lim xn→x.

isolated point definition

a point x is an isolated point if it is IN A but not a limit point

closed defintion

A set is closed if it contains all its limit points

sequential characterization for closed sets (a set is closed if and only if)

A set is closed if and only if every cauchy sequence contained in F converges to a limit that is an element of F

the union of a finite collection of closed sets is

closed

the closure of a set

The closure of a set A bar = AUL where L is the set of all A’s limit points

note: the closure of A is the smallest closed set containing A

O is open iff

O^c is closed (and vice versa)

the intersection of a arbitrary collection of closed sets is

closed

Decide if a,b, and c are open or closed (3.2.3)

3.2.3

Prove a set is closed iff every cauchy sequence contained in F converges to a value that is an element of F

3.2.5

compact official definition and characterization

official: A set is compact if for every sequence in K there exists a subsequence that converges to an element in K

characterization:A set is compact if it is closed and bounded

nested compact set property

lets sets be nested and compact Ai. then the intersection of Ai from 1 to infinity is nonempty (its basically the nested interval property)

Show that if K is compact and non-empty than both supK and infK both exist and are elements of K.

3.3.1 (hint: use lemma 1.3.8)

Prove if a set K is closed and bounded, it is compact

3.3.3 (hint: BW)

show if the following things are open or closed (3.3.4 abcd)

3.3.4

a set is perfect if it

is closed and contains no isolated points (drivel lowkey)

separated definition

two sets are separated if Aclosure intersection B is empty or if B closure intersectiong A is empty.

A set E is disconnected if

if E=AUB where A and B are separated sets

Characterization of connected sets

A set is connected iff for nonempty disjoint sets A and B theres exists a sequence in one that converges to a limit that is an element in the other

A set is connected if and only if whenever a<c<b with a,b in E, c is in E too

Prove a set is connected if and only if nonempty disjoint A and B such that E=AUB when there exists a sequence in one that converges to an element of the other.

3.4.6

functional limit definition

let e>0. let delta>0 . then if |x-c|,delta, then lim x→c f(x) = L if |f(x)-L|<e

topological funcitional limit defintion

f(x)→L if for every epsilon neighborhood of L, there is a delta neighborhood for c where if x is in c’s delta neighborhood, f(x) is in the epsilon neighborhood of L

algebraic limit theorem for functions

the same as algebraic limit theorem but its for functions

divergence criterion for functional limits

if there exits xn and yx where lim xn= lim yn but lim f(xn)=/ lim f(yn) then the limf(x) DNE

show lim x→0 of x³ = 0 using defintion of functional limits

4.2.5 b

show limx→3 1/x = 1/3 using functional limit definition

4.2.5 d

continuity definition at a point c

f(x) is continuous at a point c if for all e>0, there is a delta >0 if |x-c|<delta, then |f(x)-f (c )|<e

criterion for discontinuity

if you can find an xn such that xn→c, but f(x)/→f( c), then f is not continuous at c

algebraic continuity thm

if f(x) and g(x) are cont, then f+g is continuous, k*f is cont, f*g is continuous, and f/g is continuous if defined everywhere

compositions of continuous functions theorem

if f is continuous at c, and g is cont at f(c ), then f(g(x)) is cont at c

Prove that composite functions are continuous using the e-delta definition

4.3.3a

Show using the continuity def that if c is an isolated point of A, then f:A→R is continous at c

4.3.5 (we can set delta to be small enough to where the only point in |x-c|<delta is x=c. then f(x)=f( c) so |f(x) - f( c)|<e

4.3.8 decide if the claims are true or false

all are true

preservation of compact sets

if a function is continuous on a compact set A, then f(A) is also compact

extreme value theorem

If f is continuous on a compact K, it has a maximum and minimum

uniform continuity definition

A function is uniformly continuous if for all e>0 and there exists positive delta such that for all x,y |x-y|<delta, then |f(x)-f(y)|< e

sequential criterion for non uniform continuity

a function is not uniformly continuous if there exists xn and yn where |xn-yn|→0 , but |f(xn)-f(yn)|>=e0