- #1
jgens
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Homework Statement
Consider the open interval [itex](a,b)[/itex]. Prove that [itex]\mathrm{sup}{(a,b)} = b[/itex].
Homework Equations
N/A
The Attempt at a Solution
I'm terrible at these proofs so I would appreciate it if someone could verify (or correct) my solution.
Proof: Clearly [itex]b[/itex] is an upper bound since [itex]\forall{x} \in (a,b)[/itex] we have the strict inequality [itex]a < x < b[/itex]. Now, suppose that [itex]b[/itex] is not the least upper bound. Letting [itex]c = \mathrm{sup}{(a,b)}[/itex] there must be some real [itex]\varepsilon > 0[/itex] such that [itex]b - \varepsilon = c[/itex]. However, since [itex]\varepsilon > 0[/itex] this implies that [itex]\frac{\varepsilon}{2} > 0[/itex] and similarly that [itex]b > b - \frac{\varepsilon}{2} > b - \varepsilon = c[/itex] which contradicts the fact that [itex]c = \mathrm{sup}{(a,b)}[/itex]. This completes the proof.
I know that my proof is definitely wordy, but is it correct? Thanks for any suggestions!