Upper and lower sums - Spivak I am using the 2nd edition of the book page 218 in Calculus by Michael Spivak in the chapter of the Riemann Integral.

$$\sup \{ L(f,P) \} \leq \inf \{ U(f,P) \} $$
It is clear that both of thes numbers are between
$$L(f,P') \leq \sup \{ L(f,P) \} \leq U(f,P')$$
$$L(f,P') \leq \inf \{ U(f,P)\} \leq U(f,P'),$$
for all partitions $P'$

I really do not understand why $L(f,P') \leq \inf \{ U(f,P)\} \leq U(f,P'),$ is true. How does he justify the fact that $U(f,P) \leq U(f,P')$? He didn't state $P' \subset P$, he is claiming that this inequality is true for any partitions $P$ and $P'$. In the previous page he uses $P_1, P_2,P$ so I do not understand where $P'$ suddenly comes from
 A: Do you understand why $L(f,P') \leq \sup \{ L(f,P) \} \leq U(f,P')$ is true? 
Is the same reasoning for $L(f,P') \leq \inf \{ U(f,P)\} \leq U(f,P')$. 
The first inequality $L(f,P') \leq \inf \{ U(f,P)\}$ is true because $L(f,P') \leq U(f,P)$ for all partitions $P$ ( If $Q$ is the common refinement of $P$ and $P'$ then $L(f,P') \leq L(f,Q)\leq U(f,Q)\leq U(f,P)$). 
For the other inequality note that the infimum is taken over all partitions of your interval (let's denote it as $[a,b]$) therefore $\inf \{ U(f,P)\}=\inf \{ U(f,P):P \text{ partition of }[a,b]\}$ and this is less or equal of $U(f,P')$ since $P'$ is a partition of $[a,b]$.
A: For any partition $P$ you have $L(f,P) \le U(f,P)$. If $Q$ is a refinement of $P$, then you have $L(f,P) \le L(f,Q) \le U(f,Q) \le U(f,P)$.
It follows that if $P_1,P_2$ are two partitions, that $L(f,P_1) \le U(f,P_2)$. This is because you can always find a $Q$ that is a refinement of both $P_1,P_2$.
All of the above inequalities follow from this.
For example, since $L(f,P_1) \le U(f,P_2)$, then $L(f,P_1) \le \sup_P L(f,P) \le U(f,P_2)$.
