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Problem set 1 intro to analysis, Cheat Sheet of Mathematical Analysis

Boston College (BC)Mathematical Analysis

intro to analysis first homework

Typology: Cheat Sheet

2024/2025

Uploaded on 04/15/2025

kendall-long
kendall-long 🇺🇸

1 document

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bg1
12.8 ab
afIN IN 11but not onto
fX1 2X
NO such IN such that fxS
FIX only even numbers no odd s
Given an odd number it is not possible to
find an element such that FXodd number
NOT ONTO
bIN IN onto but not 11
fix 1it x1
X1if
where ein
f11 1412 Not 11but it is onto
Given any nein there is aXen such
That FIX EIN
pf3
pf4
pf5
pf8
pf9
pfa

Partial preview of the text

Download Problem set 1 intro to analysis and more Cheat Sheet Mathematical Analysis in PDF only on Docsity!

1 2.8 a^ b

a f IN IN^1 1 but not onto

f X1 2X

NO such IN^ such^ that^ f^ x^ S

FIX only even^ numbers^ no odd^ s

Given an^ odd^ number it^ is^ not possible to

find an element^ such^ that F X odd number

NOT ONTO

b IN IN onto but not 1 1

fix 1 it^ x^1

X 1 if

where ein

f 11 1 412 Not 1 1 but it^ is onto

Given

any

nein there is a Xen such

That FIX^ EIN

a (^) a belk

acb albts for

every

False suppose a^ b^ If you add 70 to^ b

then albte is true

a b^ _^4

Therefore this statement is^ false because

a can equal

b and^ acbte^ is^ still^ true

black if albte for

every

False use^ same^ example as^ in^

part a a can

equal

b and albte for

every

70

c asb albte for

every

70 TRUE

Assume asb

ate bte with 70

since so^ at a

acateebte

acbte for ETO

E (^) Assume (^) albte for

every

70

a set B with inf B

sup B^1 set^ with^ I^ elemen

here

sup B^ inf^

B

satisfying

infB sup B

b A finite set that contains its inf but not its

sup

This is impossible In a finite set the^

supremum

equals

its maximum The^ maximum is always

contained in^ the set^ so^ the^ supremum is^ too

c (^) bounded set Q (^) that (^) contains its (^) sup but (^) not (^) inf

A

8

Q o^

q

The

supremum

is I and contained in the

set but its intimum O is not contained in

The set^ The set is bounded between o and I

in Q

(^1) 3. a (^) Mln (^) min IN with (^) men

The suprema is 1 and the intima is^0

b 51 11m^ n M N IN

The Suprema is I and^ the intima is^1

c

n 3nti NEIN

4 1 1 4 the intima is

1 4 Yo It the styn^

The suprema is 13

d (^) m (^) imth m n IN m l^ no us In

The intima is^0

mex.net

Is (^) Mm

the y'm

The

suprema

is 1

B 1 In nein

Sup A^ SUP^

B 1

least uppler^ bounds NO (^) element in (^) B is^

larger

than 1 I isn't in^ the^ set^ B^ does^ not contain^ its (^) supremum

B does^ not^ contain^ an element

greater

than

every

element in A^ so B^ does

not iontain^ an^ element^ that^ is^ an^ upper bound^ for^ A

upper bound^

St n Not (^) upper bound S (^) In

we want to show s is an upper bound for A

suppose acA^ since^ St^ n^

is (^) an

upper

bound SUPLAIEST (^) In NEIN 116 St^ In S SUPIATES

so s^ is^ an upper bound^ for^ A

suppose there^ is^ another^ upper bound^ for^ A

called Z^ such^ that^ act

using

Archimedean property Let 70 so^ that

ne IN such that

I NLE we (^) know s In is (^) not an (^) upper bound

so S^ E^ cannot^ be an^ upper bound

so (^) s is the least among

all upper bounds

s sup A

cares X^ O^ or^0 Non (^) these are

using

the Archimedean

not in^10 n

property

if 70

so not in^ the Then^ a^

(^1) x

intersection that is^ smaller

then and

and 10 x

Thus

I (^10) n (^) does not (^) iontain

any

real (^) numbers it is^

empty

1 4.8 c

1 01 2,0 3 x

For (^) some b e^ n^ in^ then be^ nine

FOV NEIN

if belk^ then^

by the^ archimedean^ property

There exists^ an^ new^ such that ban

Then b^ n^ e

b M In intersection is^ empty