Lower sum of partition and upper sum of another partition 
Given $P_1$, $P_2$ partitions of [a, b], $$s(P_1, f) \leq S(P_2, f)$$
  where f is the function, and $$s(P, f) = \sum\limits_{k=1}^n(P_k -
 P_{k-1})m_k\text{ (lower sum)}$$ $$S(P, f) = \sum\limits_{k=1}^n(P_k -
 P_{k-1})M_k\text{ (upper sum)}$$ with $m_k = \inf\limits_{[P_{k-1},
 P_k]}f$
and $M_k = \sup\limits_{[P_{k-1}, P_k]}f$

I can't understand this theorem, how can it be possible for every partition? I think it should be
$$ P_1 \subseteq P_2$$
Or something like that...

"sandwich" another Partition $P_3:=P_1\cup P_2$ in between:
(By that notation I mean the coarsest partition being finer than $P_1$ and $P_2$)
Then by definition $s(P_3,f)\leq S(P_3,f)$ and you easily see that $s(P_1,f)\leq s(P_3,f)$ and $S(P_3,f)\leq S(P_2,f)$.
