# upper / lower integral proof

• Mar 4th 2010, 06:30 PM
Danneedshelp
upper / lower integral proof
I am trying to prove the following lemma:

For any bounded function $f$ on $[a,b]$, it is always the case that $U(f)\geq\\L(f)$, where $U(f)=inf\{U(f,P):P\in{Q}\}$ and $L(f)=sup\{L(f,P):P\in{Q}\}$. For clarification, $Q$ is the collection of all possible partitions of the interval $[a,b]$.

This seems trivial, but I am not sure how to construct a solid proof. There are a few lemmas I am temped to use, but when I try to use them, I basically just end up restating the problem statement.

Any guidance would be great,

thanks
• Mar 4th 2010, 06:53 PM
Black
It follows from the fact that if $P,R \in Q$, then $L(f,P) \le U(f,R)$.
• Mar 4th 2010, 07:59 PM
Danneedshelp
Quote:

Originally Posted by Black
It follows from the fact that if $P,R \in Q$, then $L(f,P) \le U(f,R)$.

That fact alone is proof enough...?
• Mar 4th 2010, 08:10 PM
Drexel28
Quote:

Originally Posted by Danneedshelp
That fact alone is proof enough...?

Clearly. What aspect of it are you finding difficult?
• Mar 4th 2010, 08:11 PM
Black
If $L(f,P) \le U(f,R), \, \forall P,R \in Q$, then surely $\text{sup}_{P \in Q}L(f,P) \le U(f,R) \Longrightarrow \text{sup}_{P \in Q}L(f,P) \le \text{inf}_{P \in Q} \,U(f,P)$.