- #1

mliuzzolino

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## Homework Statement

For each subset of ℝ, give its supremum and its maximum. Justify the answer.

{r [itex]\in \mathbb{Q} [/itex] : r

^{2}≤ 5}

## Homework Equations

Maximum: If an upper bound m for S is a member of S, then m is called the maximum.

Supremum: Let S be a nonempty set of ℝ. If S is bounded above, then the least upper bound of S is called its supremum.

## The Attempt at a Solution

Supremum: none, Maximum: none.

We can see that [any positive real number x such that x

^{2}≤ 5 is an upper bound. The smallest of these upper bounds is [itex]\sqrt{5}[/itex], but since [itex]\sqrt{5} \notin \mathbb{Q}[/itex], then the set has no maximum. Additionally, since [itex]\sqrt{5} \notin \mathbb{Q}[/itex] the set does not have a supremum.

I think this is correct, but I'm not exactly sure. Is there no supremum because even though the least upper bound exists, [itex]\sqrt{5}[/itex], this least upper bound is not in the set of rationals and therefore the set has no supremum?