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dec.2.x ec.I.D. fi/tsetAEhRisboundedabo v eifnuElR,s.t.
Axiomoflompletene.sn
:
1
0
1
-
1
1
2
)
aeuf orallac- A.uiscakdanupperbound.Ev ergnonemptgse.to
fr ealnumbers thatisboundedabo v elbelo wlhasaleastuppe.ir
lgreatest
loweribound.Def.AE/RisboundedbelowifnlE1R,s.t.kaVaEA,lisa
low erbound
,
ThmilNestedInterv alPr opert.gl
ForeachnelNsupposewehaveaclosedintervalDef.SE/Risthe
leastupperboundcsupre.mu
m
)
o
f
I
n
=
[
an.hn
]
=
{
XEIRlanex.hn
3
A
≤
112
if
S.t.I.ZI2ZI32-li.e.IntiEInknl.Thenf.in
≠
∅
①
:
SisanupperboundQIfuisalsoanupperboundthe.nu
≥
s
,
WhgnotinQiIn-IXEQITTEx.TT
了
N_n
⼆
粥
丰
☒
*
Wecansimilarhgdefinegreatestlowerboundc.in
f)
Whynot openinterv alsiIn. co
,
故
拓
⼆
∅
Thm
:
5
-
sup
(A)
forallao.thereisac-A.s.t.pro
ofktA-anlnEMB-ib.nl
ne
们
EIR
≤
1
1
2
5
-
《
a
I
I
…
I
I
proofa.az
bzb
,
ispss-supmand-Note.is
-9
的
,
soitcannotbean wperboundof A.hbt e.ir
allnc-NbnisanupperboundforAAxiomofhatmeansTEAS.fi
S.co
BytompletenessweknowAhasaleastupperbound.LT
spsforallaz othereisaEAS.tn
Seta-upmkteforallnc-NS-q.ca
aneaebn
ae
歞
n
在
keutobeanotherupperboundofAlwtsu.si
actan.hn?=In#,wanttoshowTheoremilArchimedeanProper
g)
BW O CSpsucs-s-u.co
①
:
GivenanyxEIRthe reis nc-NS.tn?Xlbgwagofcontradiction)
②
:
Givenangy elR Wlyno.int
NS.t.hgwhichiswesetq-s-uzoso?aEAS-9=S-Stuca
ucacontradictionlYouwanttoihinkthatxis v glargean. cl
y
isv gsmall
,