It is my purpose to examine the set of all real valued sequences with respect to certain algebraic structures. Particular consideration will be given to the set of all convergent sequences. Several of the results will be obtained by examining the properties of limit and restricting the domain of limit to the set of sequences converging to zero

Boyte, James Monroe

NC DOCKS at Appalachian State University

The University of North Carolina at Greensboro

TTh\?3"  IHVESTIGATION  OF  REAL  vALufiD  sEQUENcrsA  "®8igPr¢s®nt®d  tothe  Faculty  ®f  the  Deparfement  of RIathon`£tics±tppal€ichl&n  State  T®&ch®rs  C®11egeIn  Ptirtial Fulfillmentof the R€qulrment8  for the  Degreeifest®r  of  AmaEyJrt&nes  &tonpe®  apgiv®July  1966DEDICATIONTo  my  wife,  JenACKl`?OTi`ELEDG}ffiNTI  vlsh  to  expre88  my  81ncere  appreclatlon  to  Dr.  RalphW.  Ball  for  helping  make  the  previous  year  a  most  rewardingone  at  Appalachian  Lstate  Teachers  College.     The  success  ofthis  investigation  de8erve8  much  credit  from  the  coursestaught  by hin.AN   INVESTIGATION   OF   F.EAI,-VALUED   SEQUENCESByJane8  Monroe  BoytepettA Lu BQft2essorApproved  by:AN  IwrsHGA"ON  oF  REEL-VALt7m  sEQUE«CESAn  Abetract  of  a  The818Pre8ented  tothe  Faculty  of  the  D®p£Lctment  of prrathedmtlcsAppelachifin  State  Teachers  College|n  P&rrclal  Fulfilinentof  the  R®qulrenenii8  for  the  D®groe}haster  of  ArnoBByJabe8  Gtonzroe  Boyt®July,  1966The  thesl8  conti'.{1n8  orlgLntrll  re8ult8 with  r®8pect  to  thecot  Of  I`eaL-valued  Sequence.  and with  respect  to  8oee  of  lt88ubs®t8.    These  results  are  obtained  ty deflnlng  operatlon8on the  8et8  ty neaae  Of which  alg®bralc  structures  are  imposed.Ertyha818  18  given  to  the  ±£E±E function  producing  theorene  €'indlema8  of  inter.eat.    Among  theB€  are  two  theorene  of  ®&peclalinterest,  vi&. ,_EI_e_9_r_BE_ _i€ScReal8=HgrTliirFT-EEEEEEEEHThere  exl8t8  a  b&Blg  for  the  Set  of  all  convergentSequences  which  contains  only  one  8equence  converging  toBone  value  other  than  a.TAELE   OF  CONTENIISI.     Intztoductl®n ....................11.    Hopbe:ii:;t;:n:e?u:n:e:  :  :  :  :  :  :  :  :  :  :  :  :  :  :1 Sequence......32€:::g:::eo;fs::!::g::  :  :  :  :  :  :  :  :  :  :Diverging  to  + co or  -oo  ..........8.     Theor®mg ....,...............1I:    gn##£::=Se°fr±£;n#;:ge¥3:nt|€y:  :  :  :  :  :  :.    Sequences  form  vector  Bp€&ce  over  the  reiJs..    S`m  of  two  convergent  Se•     Product  ®f  a  ¢onstanc  i.in3Ti=    ®ruc    t     .....uences......a  sequence ....;   8£:::pe:i::if::i::§n¥3::::fe¥§:i::::: :  :  ;  ::  ;u£I!:g=°:fi£::¥3fi!igi:::; : : ; ; : :12,13.14. 93£::!3:!:-:#g!t::8`:ige#:e3:e:lg:3r:ell8.over  the  reLqls ............Proper  mfixl".il.   .   *   .........K®rn®1  of  limit--proper  maximal   .   .98fandi:*£¥#o:fn£:1:¥?e:Basis  for  range  of  `+.i  transformation  .   .Ba81s  for  spiice  of  convez`gent  8equ®nce3Spaces  of  8equenc®s  with  same  dlnenslonIll.    Bibllogramy .................. ..L5INTRODUCTIONIt  ls  my  purpose  to  ere;Limlne  the  Bet  of  all  real  v£-iluedB®qu®nces  with  respect  to  cert€iln  {:!lgebralc  Gtructur.eB.Particular con81dertitlon will  b®  glv®n  to  the  8ot  of allconvengent  Seq'uence8.    Several  of  the  re8ult8 will  be  ob-tained  by examlnlng  the  propertlee  of linlt and res€rictlngthe  domain  of  llnlt  to  the  Bet  of  8equence8  converging  to8ero,The  motlv&tl®n  for  doing  reBear'ch  ln  this  are€&  &ro3®fr`on  the  idea  that  llmlt  mlcht  be  {L±,  rinF> homouorphlan  anda linear trranafor"tlon.search  &t  Apprlachlan  f€.illed  to  dl3clo8e  lnformatlonon thl6  topic.    Therefore,  the  follenedng r®8ulta  are  regardedas  b®1ng  orlglnal :I.    Lemae  1,  2,  3  and  411.     Theoren8  2,   3,   6,   9,   10,   11,   12,  13,  Lfr,  16,   18,19,  20  and  21Ill.    Propo81tlona  1,  2  and  3.AV  IrvrsTIGATloN  OF  REAL  VALum  8EQUENcrsPofchitlon 1A  8equenc®  18  a  function wh®S®  domain  la  t,he  posltlveintegers .H®r®after  all  @®quenceB  will  h&v®  a  range  th&2.t,  1e  a  Sub-8et of the real ntrfura.P±e=f=ap4=Qlon  2£\  g®quence  {an}  convenges  to  lt3  llmlt  a,  a  real  number,1f and  only  Lr given  €> 0  then  there  exlsta  a  poaltlve  inte-ger  !S  such  that  a 2 If  inplle8   f|anrd I  <  €.E9=ftr±±±OjL£A  sequence  dlv®ng®8  1f  and  only  lf  lt  dcoe  nob  ¢onv®rge.P=±_*__te_1=t=1__OutA  8®quenc®   {fro}   dlvergee  to  +co  lf  and  only  lf  given  aae&l  n`mber  8  there  ®xLBt8  a  poslt±ve  lnt€g®r  N  Such  thatn 2 lf  fupleB  an > 8.A  g®quence  {an}  diverg®8  to  -c~  1f  and  only  lf  given  areal  ntmber  8  th®r®  ®xl8ta  .I.i  posltlve  lntBger  1€  Such  thatn 2 N  inplles  an < a.-3-Theer" IThe  linlt  of a  Convergent  8equenc$  18  unique. IP"f!Suppo8®  the  eequence   {an}  convengee  to  a,  and   {an}  Con-verg®B  eo  a'  eueh  that  a  / &'.    Put  C=  |and' I  >  0.    sincethe  €®quence   {an}  ¢oav®rgee  to  a,  there  erclste  a  p®81tLvelnt®ger  W,  ouch  that  a 2 N,  ±npllea   |anun'|  < €  .    Also  th®recoc±8t®  N£  Bush  that  n  a H£  1mp11®e   |#nun|  < £  .     Ii®t  #  ®qulthe  mELxlmm   {FJ,,N£}.     men  n  a  N  lmpll®8   |aun' I   ee   |a-dn+€inri' IS   liindnl   +   lan-*' I  <  £  +  €es €.     Thus  W®  have  a  -a'.     Thl81®  a  con€radlctlon  to  a  f  a'.    Hen€®  the  11m±t  of a  convergent8equenc®  1a  unl  ue.D®fhitlon  5I.®t   {an}  and   {bn}   be  g®qpenc®c  and  k  a  real  nutb®r,  then{:ln}  ©  {bn}   es   {€in+bn},   k{t~2n}   ee   {k.tln}    and   {an}   ©  {bn}   ca   {&n.bn}.HERE  E]The  8et  of all  Bequerice8  18  an  abell#n  gro`xp under @  .Proof :I.®t  S  be  the  act  of all  sequen¢eB.I.    €l®arly © 18 a  fun€tlon and a :  a X  S  + a.2.       {{an}   ©  {bn}  )  ©  {Cn}    S=   {din+bn}   @  {qn}    ee   { tan+ha)+Cn}   eetin+(bn+Cn)}   ca  {th}   ® {bn+¢ul    8  {in}   ®  {{bn}   @  {Cn}  }.3.     Let  €ha  a  0  ron  all  n,  and  denote  {in}   #  {On} .    Then{On}  ©  {bn}   ca   ton+bn}   e=   {ha}.j\ppLe::£hdne#€u#d°E:#::' (Now  T®rk:-4--{an}  C+`;   {an}   ee   {uen}  ©  {an}   a   {can+an}   es   {On}.{th}  ©  {bn}   =   {&n+th}   ee   {bn+Cin}   ±   {bn}  ®   {an}.Lgr=__aThe  e®t  ®f all  8equen¢es  ls  a  cormitatlve monold  underfroof :Let  S  b®  the  8.t  of all  ®equenc®s.Clearly ® 18  a  function  and \-L' :  a X  S  + S.({fin}   t:.   {bn}  }  (i.+ {Cn}    ce   tan.bn}   (.~   {Cn}   ee    {{an.h}.an}   ca{{°.n.(th.Cn)}   a   {i*n}  C,  {th.Cn}   -{an}  a   ({bn}  (i;  {Cn} ).Cl®&rly  {an}  whore  an # i  for au  n  1g  the  ld®ntlty.{tin}  rt,\  {bn}   tr~   {&n.ha}   as   {bn.an}   #   {bn} .{an} .Ca-nt:A  com&tatlve  monold  18  the  9tronge8t  €onclu81on  sincethe  Set  of  +ill  8equ®nc®g  has  dlvl8ora  or  8ere.    That  1§,91Ven   {tin}   F  {Qn}  We  Cannot  Bay   {&n}-I   eecl8t8.I_®ee=-__3The  left  {rand  right  dl8trlbutlve  Lt+lwB  of (*L,( over |.~,`, holdfor  all  8equence8.Proof :{{{1n}+{h}}  (t,``  {¢n}   #   tan+h}  r±   {¢n}   as   {€cf,tn+bn).Cn}   -{{-1n.Cn+bn.Cn}   ca   tan.Cn}  flu  {bn.Cn}   I-a   {{an}  ©  {Cn} I  ©1 (h, ©  (cn, i.2.    ¥{1cht  dl8trlbutlvo  pnoperrey  Ls  8lmllar  to  I.Thcorm 2The  3et  of all  gequen¢es  1€  a  €emltatlve  ring lnrlth  ldentlty.-5-Prcof :Len:i  1,  lerm€i  2,  and  lena  3.TThcor" 3The  Bet  of  till  Boquenc®B  18  a  vector  8pe¢®  over  the  r€alB.P"f:Let  a  b®  the  act  of  all  seqapnc®8.    fry  lerma  I,  S  18  anulbellan  gro`xp under © .    Clearly © :  R X  S  + S,    For all{an} i   {bn}   e   S  and  ii,b  6   r.  tle  have:tt„b){bn}   se   {{,'fa+b).bn}   E   {i-i.ha+b.bn}   t=   {a.bn}  ©  {b.bn}   cea o  {pn}   +  t2 o  {'8n} .a  ©  ( {an}  ®  {bn} )   8  ``'i  ©  {{an+bn} }   =   tit(an+bn}}   se{&.t'in+abn}   F-   {il.#n}  ©  {t"bn}   ffi  a ©  {t n}  ® a  ©  {bn} .(a.b)  ©  {:+n}   as   {(a.b).iln}    ra   {a{b.fro)}   ca  &  ©  {b..ant   caa © (b © (tin, '.1  ©  {+ri,n}    ee   {1.i.n}    6   {ari}  .Thcorca±The  €un  of  tilfo  conv®ngent  8®quenee8  1S  €onvergant  andthe  lLmlt  of  the  8`m  18  the  Sum of  the  llmltg.Proof :Let  € be  given  greater  than  zero.    Suppose  {#n}   converg®eto  a  and   {bn}   convezTg®B  to  b.     Then  there  exl8ts  Nt  auch  tha€n 2 N,  ±mplles   lance I  < £ dnd  th®z.a  coel3t8  N£  Bush  that  n  2 Nglmpll®8   |bnLb|  <  £  .     Let  N  ®qual  m£{]{1mm  {N,,r$2}.     Thenl{&n+bn}-(;i+b)I   E   l{an-al+{bn-b)I   i  |anca|   +   |bn-b|  < £ +  £  #€.a2ERE"  p.  35.-6-Thcoren  5If  {Th}   Converg®8  to  a,  then  for  k  e   R,  k ® {drn}   converge8to  k.a,3Proof :L®t  €> 0.    There  1g  a  posltlve  int®g®r  N  8ueh  that  n 2 N==:::nLi='i:,#=j#:2cRE<¥£®alkGn-ka|"finrfe)IThcoren 6Let  S  be  the  Set  of  {111  Bequencea  and  Sc  be  the  set  ofall  €onvengent  a®quenc®8  then  Sc  19  a  8ubBpac®  of  S.Proof :For  fill  {{-<~n}  ,{bb   e    S  end  €r.i  e   H~  we  have,   by  theorem  4.t,hat  {91H}   © {bn}   e   Sc   and  by  theorem  5  that  iL+I  ©{ .tvm}   e    fc.It  follows  that  Sc  ls  a  sub8pace  of a..ifer=qg±!±If  sequence  {Lln}   is  convergent,  then  {{in}   is  bounded.3Proof :Suppose  {an}   Conveng®8  to  a.     There  cocl8tB  N  au€h  thatn  a  1{  1mplles   Ion-.i-L|   <  I.     Thus   |an|   -|a|   S  I |an|-|a| I   SIancal  <  I.     So  n  2  r¢  LxppLle8   |an|  <   |a|+1.     I.et  8  equal  themaxlmun   {la,I,lfi2l,    .H   ,lljN.,I,|a|+l}.     Then   {an}   S  8  fortill n,3EEL.,  p.  37.4A.  I.  Ifal.cev,Fr{;1nc18Co:   .laJ.   H*FOund&tFreeman  :in aEpaHy===I=9=63S01m8ted.  2E.  g±±.I   P.   3S.g£ linear. A±E£Lt±=±i  {Sant-.-7-Thcorm  8The  product  of  two  convergent  sequences  i8  convergentand  t,h®  1inlt  of the  pr.oduct  ls  the  produce  of  the  llmlt.®Prcof :Suppose  {€ib   converg®B  to  #  and   {h}   conv®rg®s  to  b.B]r  theorem 7.  th®re  coci8t.  a > 0  Such  that  I th|   S a  for  alln.    L®t  € >  O.     Now  pick  W  >  a  Such  that  n  2 N  lmpll®8   Iba-bland  n a N  lmplle8   luln-h I  < £.    Then  n 2 N  implle8|ilnbn-*bl   ae   lanbnrabn+abn-£ibl   ee   I(£n-a)bn  +  a(bn-b) I  Sl{an-ii)bnl   +    la{Pn-b)  I   a    lbnl  lanndl   +    I:.-il  Ibm-bl   S   lan-alB   +i„ I ibn-b I  `` #  +   I:1 I::}rpe;f:;][:T S £ +  th±  .  € < € +  € B  € .Tincorem  9I.et  S  be  t,he  8®t  of  Sill  Bequence8  and  Sc  the  8et  Of  allcoevergent  sequences.    Then  Sc  ls  a  cormutatlve  subrfug  ofa with  ldentlty.Prcof :Suppose   {fln}  tina   {bn}  belong  to  Sc  then,  by  theoren  4and  theoren  5,   {tin}~{bn}   ee   {&n}ex-bn}   e  Sc.     B]r  theorem  8,{inLn} ©  {bn}  ee   tan.bn}   e  Sc.     It  follcws  that  Sc  ls  a  Subrdngof  S.?    Colrmtcltivlty  is  lnherlted,  and   {+3n}  ee  {1n}  1s  theidentity.¥,ie~o_rHusliet  3c  be  the  Set  Of  1'111  Convergent  B®quenc®8.     Then€ERE.|8ost:LV?L#e8.St.]eaBfumffioductionasyT"3# ffi€.ire --8-limit  18  a  ring  homomorphham mapping  the  Bet  of  all  conver-gent  sequences  onto  the  real  numbers.Proof :}3y  theor+em  1,  limit  18  a  function*    The  fact  that  thereal8  are  complete  lmpll®8  linlt:  Sc  + Ti.    Clearly llmltmaps  Sc  onto  the  real8.    Let    tin  ,  bn    belong    to  Sc,  then  bytheorem  4,   (  {&n} ©  {bn})llmlt  ac   ( {€in})limit  +   ( {bn})limit,and  by  theorem  8,   {{i;n}©{bn})limit  =   ({an})limit   .   { {bn}}1imit.Theorm 11LI___  I__    I__T_   ___   I      _    i___Limit  ls  ,.i  linear.  transfor"!tion mapping  the  set  of ;illConv®ngent  8equence8  onto  the  real  n`mbers.Proof :Let  Sc  be  the  Bet  of  all  convene.ent  sequences.    Suppose{an},  ton}  belong  to  Sc  &nd  a,b  belong  to  r.,   then   (a{€in} ®b{bn})limit  a   ({a.din} 6  {b.bn})limit  g  ({a.an})1inlt  +({b.bn})limit  =   (a © {an})limit  +   {b©  {bn})1inlt  esa   .   {({iin})limit)  +  b   .   (({bn}}1imit).     Limit  is  a  functionby  theorem  1.    Thus  llmlt  belongs  to   ¥(Sc,R}.    Clearly  limitmaps  onto  the  real  numbers.Theorem  12rieai8 = rfuProofBy  theoren  10,  1inlt  18  a  ring  houomorphlam mapping  Sconto  the  real8.    By  fundamental  theorem  of  ring  homomorphlsm,ScWe  have  her iinit    =  {S¢)limit  g  R.88RE.,  p.  90.-9-Theorm 13The  8®t  of all  3eqtrence8  18  a  linear algebra  over R.Prcof :Let  a  be  the  eec  of all  €equenceB.    By  theorem  3,  S  lsa  vector  Sp&¢®  over  the  I+ealB,  and  by lena  2,  ©  :  a  X  S  + Sand  ls  aBeoelatlve.    Let   {;in},{bn},{cn}  belong  to  a  and  ti,hbelong  to  r.     Then   {an}  ®  (:i{cn}  ©b{bn}}  #   {an}  ©    ({acn}  6{bha}}   se   {an}  ®  {{&Cn+bin}}   c   {an{¢1cn}   +   an(bbn)}   ca{.i(ancn}   +  b(anbh}}   g   {`1({|ncn)}  ®  {b{anbe}}   -a  ©  {anon}  ©b ©  {anth}  ee  a ©  { {an}. {Cn} )  ® b ©  { {an}  ©  {bn} ).     $1mllL#rly.{a{an}  © b{h} )  ©  {¢n}   €  a ( {an}  ©  {Cn} }  © b{ {bn}  ©  {Cn} LEiRT+EHtt+  i+     E]EThe  8®t  Of  all  conveng®nt  gequence6  1g  a  linear  algebraover the  realB.Prmf:Let  Sc  b®  the  cot  of  aill  convergent  sequences.    Bytheor.en  6,  Sc  LB  a  8ubspac®  Of  the  eec  of  till  8equene®s.B]r  theoz.em  9, ®  :  Sc  X  fc  +  S¢  and  18  a8Boclcktlve.    Theblllnear property ls inherited.i.tHL®t  R  b®  a  eonEzmt:..¥.tlve  ring with  ldentlty  I  ¢ 0.     Anideal  I  ln  !118  proper tryaxlnal  lf  and  only  lf i 18  a  field.9EBfflRErfeThe  k®m®l  of lfmlt  18  proper quLrful  1n  the  set  of  allconvergent  &®qu®ne®s.9ERE„  p.  126.-10-Proof :Let  Sc  b®  the  a®t  of  all  convergent  8®quence8.    Thekernel  of 1±mlt  of  So  18  an  lde&l.'°  P5r  theorem  12,Scker(|hit)1a  a  fl®ld  81nc€  the  rleal8  1e  a  fl®1d.    W®  have  Sc  a  com-          \\"tatlv®  ring wleh  ldentlty.    Thus  by  th®oron 15,  the  th®orenfollows ,EbeERELet  V  {md {vAJ  be  vector  8Face8  with  f  a  llncar  tran8for-"atlon  mapping  V  into ``f.    Lot  C  ra  {v* }<ej^t  b®  a  baol8  for  the"111  8Pac®  Of  I.     If  n  in   {V*}*eTj  18  a  8®t  of  Y®Ctorg  Suchthat  Cub  18  A  ba®1B  for  Y  with  CrlD  =  ¢,   then  {(V*)f}*g3  i8a  basl3  for the  range  af f .Proof :Let v  belong  to  the  range  of  f .    Th®r®  exlst8  ri  v  e  VnSuch  that   {V)f  #  W.     Now  V  #  ±ca£L'`j|VAL.     Then  W  ae   (V)f  es(i:£Lalv<i)f  r-1:£hL[i(V<£)f  g *±£/A`rti(V<£}fi   Since  <i e    A  inpllega{V*L)f  ==  0.     SO   {{V«}f}«eB     8P€in8  the  range  Of  f.     I.et  1:L&i{V*i)fa  0 with  *i  e  „    Let  v  a  ±g[fiiY<±.    ¥hen  (V}f  gr (±gLfiiv*±)  f  ¥ni3|ai(v<i'f # 0.So  v  is  ln  the  null  space  of f .    Thus  thereinnexl8ts  *j   e  ,i`i  Such  that  V  g  j£]bjv<±  ca  ±£Lrjiv<±    With  *±  e  8.innHence  jE[bjv*j  +  ±£L-{iiv^±  as  0  implies  ai  ee  0  for  i"  i.    Sotore..  p.  90.-11-{{V*)f}* efi  18  a  liriearly  independent  Set.a-eat :The  linear  tran8fol"tion  f,  restricted  to  {}`!<}*et7.  isone  to  one.Th®oren  18There  exlst8  a  b;:isls  for the  cot  of all  ConvergentBoquences  which  cone+r§1n8  only  one  sequence  converging  to8ome  viilue  other  than  0.Prcof :Consider  the  b€igls   {V*}<eAUE  ln  th®ol.lan  17.     W®  havechown  th&t  {(V<}f}*eB  ls  `.;  ha818  for  the  range.     Now  con8i-d®ring  f  E  linlt  find  {v<LetuB  trj  ba81B  for  the  Bet  of  Conver-gent  Bequence©,  *®  have   {lv*)11mlt}<eE  19  a  bids18  for  thereels.    The  ba6ls  for  the  reaL8  con81stc  of  one  number rf  0.Thus  th®ro  exists  &  Blngle  N*  e   {S;<}<eB  since  limit  restrictedto   {NJ*eB  18  One  to  one.     Hence  thel.®  ®xlBtB  only  oneS©qpence  ln   {VJ<ej.LUB  which  Converg®B  to  sonethlng  otherthan  zero.Counent :A  sequence  Converges  lf  and  only  lf  lt  eon  be writtenag  a  flnlte  lln®cir  conbin``itlon  of  conveng®nt  8©quenceB withat  most  one  8®quence  ln  the  €ombin{`itlon  not  convenging  toZero ,H9=P9ffi[±ELThe  1d¢ntlty  86quence  would  8grve  ae  the  Sequence  lntheorem 18 not  convenglng  to  zero.-12-Defhition 6The  cardlnallty  of  the  natur€Ll  nunberg  18  No  and  thecardin&11ty  of  the  real  mmb®ro  is  I .Thooren 19The  dfronslon  of  the  null  @p&€®  of  llmlt  lc  at  least  No.Prcof :Let  A  b®  the  set  of  all  6equenc®s  h&vlng  the  number  11n  e3cactly  one  efitry  and  =ero8  elaenthm.e.    Clearly  A  18  allnearly  lnd¢pend®nt  Subset  of  the  Set  of all  sequen€e8conv8rglng  to  zero  and  the  c&rdlnallty  of  A  18  No.±9ap= '±_I-et  S  be  the  space  of all  sequences,  then  a  has  car-dinallty  I .Proof :The  caridlntlllty ®f  the  roals  ls  I .    Let  a  bo  the  setof  all  gequenc6a,  then  S  #  F X   R X   ri   ...   {No  times).    ThusI S|    ee   I .   I .   I  ...   (Ito  tha8}   #  I fu.a  g   (2NO}HO  a  2REO"o  82E #  I .Theorem  20Iiet  S  be  the  apace  Of  €¥11  Sequences,  Sc  the  apace  ofall  co»vgrg®nt  ©®qu®nces,  and  SN  the  null  apace  of llmlt,then  the  cardlnallty for  the  ba81a  of  each  L8  at  let.riot  Iloand at ro8t  I .Proof :Apply  theorem 19,  end  lena  4.-13-Thcore8  21Ii8t  SN  be  the  null  Bp8ce  Of llmlt  and  Sc  the  eec  of  allconvergent  B®quenc®s,  then  the  dinen81on  of  Sw  18  equal  tothe  dinenBlon  Of  Sc.Pruef :If  dlnransLon  of  SN  -  No  thou  by  applying  theorem 18,No  +  1  -No  18  the  dlnen81on  for  Sc.     If  d±nenalon  of  Sat  ts  Ithen  I +  I  rs  [1e  the  dlnenslon for Sc.tiriE+*HE!HH      EThe  dlmen&ion  for  the  null  sprLc®  of  llmlt  18  }Jo.Procogltlon 3The  d±meuslon  for  the  apace  of  all  8equenc®8  1s  I  ,BIBLIOGRAPHYLBRARXAp.peladilan   State   toecfiae   tirrl®g!-Boone'    N.    f`.BIBLIOGRAPHYBrm®3 :  %:1#::dinEandngEfgs::S±±£gfs Abstract ±±fis±=a.    BOston :Mar 'Ct#'Ht.F::enEaREgtl#,±±B%a: Algebra.    Sap Francl8co :Olm8t®d,   John  3'rl.   IZ.Appletonutentnyermedla New  York:

An Investigation Of Real Valued Sequences

http://libres.uncg.edu/ir/asu/f/Boyte_James_1966_Thesis_Investigation Of Real Valued Sequences.pdf

An Investigation Of Real Valued Sequences

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