Let $\displaystyle [a,b] \rightarrow{}\mathbb{R}$ a bounded fuction.

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

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

Thanks a lot

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- Dec 11th 2009, 03:15 AMosodudPartitions
Let $\displaystyle [a,b] \rightarrow{}\mathbb{R}$ a bounded fuction.

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

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

Thanks a lot - Dec 11th 2009, 06:46 AMPlato
To do this you need to understand a refinement of a partition.

Prove that if $\displaystyle Q' $ is a refinement of the partition $\displaystyle Q$ then

$\displaystyle L(Q,f) \leqslant L(Q',f) \leqslant U(Q',f) \leqslant U(Q,f) $.

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