實數系

實數系. Ordered sets. completeness(完備性). 實數上有界子集合的sup 與inf. 代數結構. Ordered field. Archimedean property ... ¹ê¼Æ¨t Orderedsets completeness(§¹³Æ©Ê) ¹ê¼Æ¤W¦³¬É¤l¶°¦Xªºsup»Pinf ¥N¼Æµ²ºc Orderedfield Archimedeanproperty ³o­Ó¥DÃD¬O½Í¹ê¼Æ¶°¦XR¨ã¦³ªº°t³Æ,©Î»¡¬O¯S©Ê.Á`¦@¤À¤T¤è­±:(1)order(2)§¹³Æ©Ê(completeness)(3)¥N¼Æµ²ºc.(¾ú¥v°Ñ¦ÒI;&nbsp¾ú¥v°Ñ¦ÒII;&nbsp¾ú¥v°Ñ¦ÒIII) Orderedsets ©w¸q:order. ¥OS¬°¤@¶°¦X," (1)¹ï¥ô·Nx,yS,x,«h´N¤£¬OEªº¤U¬É. «hºÙ¬°Eªº³Ì¤j¤U¬É(greatestlowerbound©Îinfimum). ²Å¸¹:=infE. Q:Á|¨Ò. top R¤W¦³¬É«DªÅ¤l¶°¦Xªºsup»Pinf. ©w²z.³]ER,E¦³¤W¬É.«h (i)­YsupE¦s¦b(R),«h¬O°ß¤@ªº. (ii)R¬°Eªº¤@­Ó¤W¬É.«h =supE ¹ïEªº¥ô¤@¤W¬ÉM,M¦¨¥ß. ¹ï¥ô·N>0,¦s¦bx0E,º¡¨¬x0>- ¦bE¤¤¦s¦bº¥¼W¼Æ¦C{xn},¨Ï±olimnxn=. ¨Ò.E={xQ|x2<2}.«h¦bR¤¤¦s¦bEªºsupremum(=21/2).¦ý¦bQ¤¤¤£¦s¦bEªºsupremum. top completeness ©w¸q:leastupperboundproperty ³]S¬°¤@orderedset,SºÙ¬°¦³leastupperboundproperty¹ïS¤¤¥ô¤@«DªÅ¦³¤W¬É¤l¶°E,¦s¦bS,¨Ï±o=supE. ¨Ò.Q,N³£¬Oorderedset.Q,N¨S¦³leastupperboundproperty. Completenessaxiom(§¹³Æ¤½³])ofR:R¦³leastupperboundproperty. ³]S¬°¤@orderedset,«hS¦³leastupperboundpropertyS¦³greatestlowerboundproperty(©w¸q¦P¤W). top ¥N¼Æµ²ºc ¹ê¼Æªº¥N¼Æµ²ºc¬O½Í¨ä¤¸¯À¶¡¤§¹Bºâ¤Î¬ÛÃö©Ê½è. ¹Bºâ:"¥[:+"»P"­¼:". µ²ºc:¹ê¼Æ¶°¦X¦b¦¹¤G¹Bºâ¤U§Î¦¨"Åé"(field). top Orderedfield F¬°¤@orderedfieldF¬O¤@­Óorderedset¤]¬O¤@­Ófield.¥B¹ïx,y,zF,º¡¨¬ (1)­Yy0(0¬°F¤¤ªº¥[ªk³æ¦ì¤¸¯À),y>0,«hxy>0. top ­Yx>0,ºÙx¬°¥¿(positive).­Yx<0,ºÙx¬°­t(negative). ¨Ò.R,Q³£¬Oorderedfield. orderedfieldº¡¨¬ªº©Ê½è. (i)­Yx>0,«h-x<0.(©Î:­Yx<0,«h-x>0). (ii)­Yx>0,yxz. (iv)­Yx0,«hx2>0. (v)­Y00,«h¦s¦b¥¿¾ã¼Æn,¨Ï±onx>y. ¥Ñªü°ò¦Ì±o©Ê½è¥iµý±o:(Q¦bR¤¤¦³¸Y±K©Ê) ³]x,yR,x