Intro to Real Analysis: Supremum

  • #1

Homework Statement


Find the supremum of E=(0,1)


Homework Equations





The Attempt at a Solution


By definition of open interval, x<1 for all x in E. So 1 is an upper bound. Let M be any upper bound. We must show [tex]1<=M[/tex]. Can I just say that any upper bound of M must be greater than or equal to one based on the definition of open interval again?

I'm just not sure if that last line is completely correct. Thanks.
 

Answers and Replies

  • #2
155
0
Use your theorems.

s is the least upperbound if for all [itex]\epsilon > 0[/itex], there exists an [itex]a \in (0, 1)[/itex] satisfying [itex]s - \epsilon < a[/itex].
 
  • #3
164
1
Since you are only required to find the sup(E) you can just state it.

Hint: sup(E) ∉ E.
 

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