# Thread: Greatest lower bounds and least upper bounds

1. ## Greatest lower bounds and least upper bounds

12

2. Originally Posted by mthrcks300
Let A be a partially ordered set. Suppose A is a subset of B and B is a subset of C. Assuming that all the least upper bounds and greatest lower bounds exist, prove that: glb(B)≤glb(A).
I start you off. If $\displaystyle x$ is the greatest lower bound for $\displaystyle B$ then $\displaystyle x\leq b$ for all $\displaystyle b\in B$. But then $\displaystyle x\leq a$ for all $\displaystyle a\in A$ because $\displaystyle A\subseteq B$. This means that $\displaystyle x\leq y$ where $\displaystyle y$ is greatest lower bound for $\displaystyle A$ because $\displaystyle x$ is a lower bound for $\displaystyle A$ and $\displaystyle y$ is the greatest lower bound for $\displaystyle A$.

3. You start by saying that x is the greatest lower bound for B, but then you say that y is the greatest lower bound for B. I'm confused.

4. I made a mistake. Does it make more sense now?