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Upperbounds problem

  1. Oct 3, 2011 #1
    Let S[itex]\subseteq[/itex] R be non empty. Show that u[itex]\in[/itex]R is an upperbound of S iff the conditions t [itex]\in[/itex]R and t>u implies t[itex]\notin[/itex]S.




    Let S[itex]\subseteq[/itex] R be non empty. Assume u[itex]\in[/itex]R is an upperbound of S. Then for all x[itex]\in[/itex]S x[itex]\leq[/itex]u. Then choose a t[itex]\in[/itex]R such that t>u. Since t>u this implies that t[itex]\notin[/itex]S since u=SupS

    Let S[itex]\subseteq[/itex] R be non empty and t [itex]\in[/itex]R and t>u implies t[itex]\notin[/itex]S for some u [itex]\in[/itex]R. So either u[itex]\in[/itex]S or u[itex]\notin[/itex]S. If u[itex]\notin[/itex]S then u is an upper bound of S. So consider u[itex]\in[/itex]S and let u be the largest element is S such that u<t. This implies that us us the largest element in S since t[itex]\notin[/itex]S.



    Is this right?
     
  2. jcsd
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