# MATH 32BH Study Guide - Winter 2019, Comprehensive Midterm Notes - Xz, Xtc, Xd-Picture Card

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MATH 32BH

MATH 32134

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01/09/2019

Lemmas Lcf ,p)EUH ,p)

Definition LE Racompact interval ,Papartition of B,

than apartition P'of Iis arefinement if P's P

BER "

is abox ,

PePix Pax ---xPn apartition of B,than

P' =P,

'xBe'x---XPn '

apartition of Bis

arefinementofp-ifp.is

Pi for each i-4,2,

...,n

The point is each P-sub box is partitioned by P'

-subbox

p#P' MIEL

Lemme BER "

abox f:B→Rbounded ,P,P'partitions of B

P'arefinement of P.Then ,

Klf

,P)ELcf ,P')EUH .p')EUH ,p)

Hoot

:Begin by noting if J'EJmy Cft Emp HI

Fix aP-sub box J

msHIWUJt-m-HIEyol.LT')=¥msHI volts ')E,ImjHI roll J')

Summing over all P-sub boxes J,

Uf .p)=Imy A)roll J)EE my 'HI roll J't)=L Lf,p

'I

JFsubbox JR subbox (EET

Argument for Uis similar I'P' -subbox

(T'ET Ms Hts Me 'Htt

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Definition (Common Refinement )

BERN abox P=P,x.-xPn P'=P ,

'

x.-.XP '

npartitions

tph.IT#xti79Ipontu9snY

'

is III. HE

Lemma_ BE Rn ,

f:B→Ras usual P,P'partition of B

Then Lff ,Pl

sUH ,P')

Boot Let P"be the common refinement of P,P'

Lff ,P)eLI f,P")§

UH ,P")EUH ,P')

first Lemma sectored Lemma

Definition

BER "box ,f:B→Rbounded

LfBf =sup ILl f,p):Papartition of B)

UfBf =int IUlf ,p):Papartition of

Bl'integrable

If Lf Bt -

-U!Bt ,then we say fis lRiemann )(over

B)

Notes :

①Lf Bt always exists since any Ulf ,pl is an upper bound for

{Lff ,p):Papartition of B)

Similarly ,Uf Bt always exists

②LIB fell fest

This follows from :

Lemme Let U,LER set .l

su for every IEL,

hell .Then ,sup

LE infill

Boot Each be Lis alower bound for It .So ,LE int th

for each be L,i. e.intel is an upper bound for L

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