# Thread: working with upper bounds

1. ## working with upper bounds

Suppose that b is an upper bound for a set S of real numbers. Prove that if b E S, then b = lub S.

2. Originally Posted by WartonMorton
Suppose that b is an upper bound for a set S of real numbers. Prove that if b E S, then b = lub S.
$S\leq b$ $\forall$ $s\in S.$

Assume $b \in S.$

Set $m=\sup{S}$ so, $S\leq b \leq m$ by assumption.

Then, since $b$ is an upper bound, and $m$ is the $l.u.b.$ $S\leq m \leq b.$

Therefore $b \in S \Rightarrow b= m =\sup{S}$

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