Let's say $\displaystyle S \cap T \not = \varnothing$ and $\displaystyle s \leq t, \forall s \in S, \forall t \in T$. I think this can only be true for two cases:

Case 1: $\displaystyle s \leq t, \forall s \in S, \forall t \in T/S$

Case 2: $\displaystyle s \leq t, \forall s \in S/T, \forall t \in T$

If so inf$\displaystyle (T)$ $\displaystyle \not <$ sup$\displaystyle (S)$

Then sup$\displaystyle (S)$$\displaystyle \leq$ inf$\displaystyle (T)$ holds for $\displaystyle S \cap T =\varnothing$ and $\displaystyle S \cap \not = \varnothing$

Yah?