Proving existence of utility function
Το take this one out of the Unanswered queue:

The completeness property of the preference relation, implies that for all non-negative $t$ we will be able to form and declare the preference relation. Hence

$$A \cup B = \mathbb{R_+}$$

By monotonicity we have $B=[0, \overline{t}],\;\; A=[\underline{t}, \infty)$.

Ad absurdum, assume that $\overline{t} &lt; \underline{t}$. Then there exists an open interval $(\overline{t},\underline{t})$ that does not belong to the union of $A$ and $B$. But then we arrive at $A \cup B \neq \mathbb{R_+}$ which contradicts the implications of the completeness property. 

So we conclude that $\overline{t} \geq \underline{t}$, which implies that the intersection $A \cap B$ is non-empty (even if it may contain a single element, when $\overline{t} = \underline{t}$).