# Partitions

Printable View

• December 11th 2009, 03:15 AM
osodud
Partitions
Let $[a,b] \rightarrow{}\mathbb{R}$ a bounded fuction.

Prove that if $P_1$ and $P_2$ are partitions of $[a,b]$ then:

$L(P_1,f) \leq{U(P_2,f)}$

Thanks a lot
• December 11th 2009, 06:46 AM
Plato
Quote:

Originally Posted by osodud
Let $[a,b] \rightarrow{}\mathbb{R}$ a bounded fuction.
Prove that if $P_1$ and $P_2$ are partitions of $[a,b]$ then: $L(P_1,f) \leq{U(P_2,f)}$

To do this you need to understand a refinement of a partition.
Prove that if $Q'$ is a refinement of the partition $Q$ then
$L(Q,f) \leqslant L(Q',f) \leqslant U(Q',f) \leqslant U(Q,f)$.

Then to do your problem find a refinement of $P_1\cup P_2$.