I was wondering, given a partition P and a refinement of P, namely Q, and a bounded function f:[a,b] --> R,

how can I prove that U(Q,f)<=U(P,f)?

Here is what I have, Im hoping someone can confirm that its right or let me know where I went wrong??...

Proof:

Let .

Suppose P={ } and let Q be a refinement of P such that Q contains one additional point of [a,b] say c where .

let =sup{f(x)| ]} and

let =sup{f(x)| ]}.

let =max{ }. [should this be min??]

Then U(P,f)=

=

=U(Q,f).