复旦数学分析教案01实数系的连续性

121Newton,LeibnizCauchy,Weierstrass2DedkindBolzano-WeierstrassCauchyCantor3DedkindDedkind431⇒⇒⇒,,2S∃∈ξS∀∈xSx≤ξξSξ=maxS,∃∈ηS∀∈xSx≥ηηSη=minSSSmaxSminSmaxSminSSA=≥{|}xx0=0minAB=≤{|}xx01SxxSxSx∈∃∈∀':',SUSUU?UββS,β=supSβaβS∀∈xSx≤βbβS∀ε0∃∈xS,x−βε4xx=x+(x)xx(x)xx=34.,[]x=3,().x=04x=−27.,[]x=−3,().x=03(x)(x)=,012.aaanaa0129(an12x)0()0000012.aaapap≠0199912.()aaap−(x)Sa=naaaa210.0+0x,=(012.aaanx),xS∈5SS(α0α0S)S0=∈={|[]}xxSxα0∀S0x∈Sx∉S0xα0S0α1S1=∈{|}xxSx01αS1∀x∈Sx∉S1xα0+0.α1nSn−1αnSn=∈−{|}xxSxnnn1α∀Snx∈SxSn∉xα012n+0.αααS⊃S0⊃S1⊃⊃Sn⊃α,012nαααα0∈Z0,1,2,9},αk∈∀∈kNβ=α0+0.α1α2αnS6βSan∀∈xS00≥xSn∉0,n≥0xSn∈xSn∉0xα0+0.α1α2αn0β()xSn∈∀∈nN∪{}0Snxβx=β∀∈xSxββS(b)∀ε0n01100nεxSn00∈βx0n0β−x0≤1100nεx0−βεβεβ−S7TxxQxx=∈{|}022TQTQsupT=nmmmnN∈n1()nm232(a)1()nm22222−=nmt01trnmt=6,nmr+0,nmrQ+∈()nmrrnmrt+−=+−22220nmrT+∈nmT(b)2()nm23nmt222−=,01trnmt=6nmr−0,nmr−∈Q()nmrrnmrt−−=−+22220nmr−TnmTTQ412“”,2SxxSxSx∈∃∈∀':',βSβ≤∈∀xSx:βSεβε−∈∃∀xSx:,0β{}2,:2∈=xQxxTQQxT∈=0sup0rTrx∈+00xTrxxTx−≤∈∀0,0x3