I've been attempting to solve the following problem for hours on end but there is one crucial part that I just can't seem to get right:
Let $( S, \geq )$ be a partially ordered subset of $( T, \geq )$. Prove that if $t=inf_T E$ , for some $E \subseteq S$, then
$$t=inf_T ( [t》\cap S )$$
Where [t》= {$x\in T$|$x \geq t$}.
Now, I've managed to show that IF $inf_T ( [t》\cap S )$ exists, then it equals t, however I am lost as to how to prove the existence of this infimum.
I'd really appreciate some hints, tips, solutions etc.
Thanks, Marius.
What if we just follow the definitions? First let me restate your question in normal notation, because your notation makes my head spin.
Suppose that $(T,\le)$ is a partially ordered set, $E\subseteq S\subseteq T,$ and $t$ is the greatest lower bound of $E$ (in the partially ordered set $T$); we want to show that $t$ is also the greatest lower bound of $\{x\in S:x\ge t\}.$
Clearly $t$ is a lower bound of $\{x\in S:x\ge t\}.$ Let $u$ be any lower bound of $\{x\in S:x\ge t\};$ we have to show that $u\le t.$ Since $u$ is a lower bound of $\{x\in S:x\ge t\},$ and since $E\subseteq\{x\in S:x\ge t\},$ it follows that $u$ is also a lower bound of $E.$ Since $u$ is a lower bound of $E,$ while $t$ is the greatest lower bound of $E,$ it follows that $u\le t.$