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Math Help - greatest least bound and least upper bound proof

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    greatest least bound and least upper bound proof

    I've been struggling with this one.

    Let A be a partially ordered set. Suppose X⊆Y⊆A.

    1. Assuming that all the least upper bounds and greatest lower bounds exist, prove that glb(Y) ≤ glb(X) ≤ lub(X) ≤ lub (Y)

    2. Find two subsets X and Y of Real Numbers for which X is a proper subset of Y and yet glb(Y) = glb(X) and lub(X) = lub(y)
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by tae1466 View Post
    I've been struggling with this one.

    Let A be a partially ordered set. Suppose X⊆Y⊆A.

    1. Assuming that all the least upper bounds and greatest lower bounds exist, prove that glb(Y) ≤ glb(X) ≤ lub(X) ≤ lub (Y)

    2. Find two subsets X and Y of Real Numbers for which X is a proper subset of Y and yet glb(Y) = glb(X) and lub(X) = lub(y)
    1. I'll do part of this. See if you can do the rest.

    Problem: Let X\subseteq Y. Prove that \sup\left(X\right)\le\sup\left(Y\right)

    Proof: We know that since X\subseteq Y that x\in X\implies x\in Y. But \forall x\in Y\quad x\le\sup\left(Y\right), so \forall x\in X\quad x\le \sup\left(Y\right). Therefore \sup\left(Y\right) is an upper bound for X. And by definition \sup\left(X\right) is at least as small as any upper bound of X. The conclusion follows.

    2. What about X=(0,1) and Y=[0,1]?
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