Thread: partition/darboux sums question

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).