- #1

Samuelb88

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## Homework Statement

Does [itex][0,1] \times [0,1][/itex] in the dictionary order have the least upper bound property?

## Homework Equations

**Dictionary Order.**(on [itex]\mathbb{R}^2[/itex]) Let [itex]x , y \in \mathbb{R}^2[/itex] such that [itex]x=(x_1 , x_2)[/itex] and [itex]y = (y_1 , y_2)[/itex]. We say that [itex]x < y[/itex] if [itex]x_1 < y_1[/itex], or if [itex]x_1 = y_1[/itex] and [itex]x_2 < y_2[/itex].

**Def'n.**An ordered set [itex]A[/itex] is said to have the least upper bound property if every nonempty subset [itex]A_0 \subseteq A[/itex] that is bounded has a least upper bound.

Assume that the real line has the least upper bound property.

## The Attempt at a Solution

I'm not sure if I am proving this correctly. Here's my proof.

I want to show that every subset [itex]A_0 \subseteq [0,1] \times [0,1][/itex] that is nonempty and bounded has the lub property. Suppose that [itex]\mathbb{R}[/itex] has the lub property. Let [itex]A_0[/itex] be an nonempty subset of [itex][0,1] \times [0,1][/itex]. Since [itex][0,1] \times [0,1][/itex] is bounded, it follows that every subset of [itex][0,1] \times [0,1][/itex] is bounded. We will consider two forms of [itex]A_0[/itex], that is, when either [itex]A_0 = [i,j] \times [k, \ell][/itex], or [itex]A_0 = (i,j) \times (k, \ell)[/itex], where [itex]0 \leq i, j, k, \ell \leq 1[/itex].

If [itex]A_0 = [i,j] \times [k, \ell][/itex], then [itex]\forall x \in A_0[/itex], we can always find a least upper bound, say [itex]y[/itex] by letting [itex]y=x[/itex]. So that case is settled.

Instead, suppose that [itex]A_0 = (i,j) \times (k, \ell)[/itex]. Then [itex]\forall x \in A_0[/itex], we can still always find a least upper bound which we will again call [itex]y[/itex] such that [itex]y = (y_1 , y_2)[/itex] by letting [itex]y_1 = j[/itex] and [itex]y_2 = k[/itex].

In a similar manner, we can show that subsets that have both closed and open ends (e.g. [itex](i,j] \times (k, \ell][/itex]) always have a least upper bound.

Therefore I have shown that every subset of [itex][0,1] \times [0,1][/itex] that is nonempty and bounded has the lub property and therefore the set [itex][0,1] \times [0,1][/itex] has the lub property.

How does this look?