Math Help - Upper Bounds and Supremums

1. Upper Bounds and Supremums

Suppose that A is contained in the set of all real numbers and is bounded above. Prove that if A contains one of its upper bounds, then this upper bound is sup A.

It seems like it should be a simple problem, but I'm lost.

2. Re: Upper Bounds and Supremums

Originally Posted by lovesmath
Suppose that A is contained in the set of all real numbers and is bounded above. Prove that if A contains one of its upper bounds, then this upper bound is sup A.
Suppose that $a \in A\; \wedge \;\left( {\forall x} \right)\left[ {x \in A \Rightarrow x \leqslant a} \right]$.

Now prove that $a=\sup(A)$.

Hint: If you assume that $a<\sup(A)$ there is an intermediate contradiction.