# Thread: upper / lower integral proof

1. ## upper / lower integral proof

I am trying to prove the following lemma:

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

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

2. It follows from the fact that if $\displaystyle P,R \in Q$, then $\displaystyle L(f,P) \le U(f,R)$.

3. Originally Posted by Black
It follows from the fact that if $\displaystyle P,R \in Q$, then $\displaystyle L(f,P) \le U(f,R)$.
That fact alone is proof enough...?

4. Originally Posted by Danneedshelp
That fact alone is proof enough...?
Clearly. What aspect of it are you finding difficult?

5. If $\displaystyle L(f,P) \le U(f,R), \, \forall P,R \in Q$, then surely $\displaystyle \text{sup}_{P \in Q}L(f,P) \le U(f,R) \Longrightarrow \text{sup}_{P \in Q}L(f,P) \le \text{inf}_{P \in Q} \,U(f,P)$.