The problem is: Let $\dot{P}$ be a tagged partition on $[0,3]$. Show that the union $U_{1}$ of all subintervals in $\dot{P}$ with tags in $[0,1]$ satisfies $[0,1-\Vert{\dot{P}}\Vert]\subseteq U_{1} \subseteq [0,1+\Vert{\dot{P}}\Vert]$.
My question is the next: If I take the partition $P=(0,0.5,0.7,3)$ and tags $t_{1}=0.3,t_{2}=0.6,t_{3}=0.8$, then $\Vert{\dot{P}}\Vert=2.3$. ¿What happend with $[0,1-\Vert{\dot{P}}\Vert]\subseteq U_{1}$? I´ll have $[0,1-2.3]\subseteq U_{1}$ and it doesn't have sense.
This problem is from the book "Introduction to Real Analysis" by Robert Bartle. (Section 7.1, exercise 4(a))
Note that $[0,1-2.3] = [0,-1.3] = \{ x\in \mathbb{R}\, | \, x \geqslant 0 \,\text{and}\, x \leqslant -1.3\} = \emptyset \subseteq U_1$, since the empty set is a subset of any set. It is true vacuously that every element of the empty set belongs to $U_1$ since there are no elements.