Proof that Upper Riemann Integral > Lower Riemann Integral...?

heyyy guys....

im new at this level of maths so i would appreciate a lil bit of help here and there :) hehe

i want to show that

lower riemann integral $\displaystyle \int^b_a f(x)dx \leq $ upper riemann integral $\displaystyle \int^b_a f(x)dx $

i have worked this out so far but im not sure if its correct: please help me on my solution.

proof:

if P,Q are partitions of P[a,b] (i.e P,Q $\displaystyle \epsilon$ P[a,b] then

L(f,P) $\displaystyle \leq $ U(f,Q)

If we let R:=P U Q, be a common refinement of P & Q then we see that:

L(f,P) $\displaystyle \leq $ L(f,R) &

U(f,R) $\displaystyle \leq $ U(f,Q)

We know L(f,R) $\displaystyle \leq $ U(f,R) so now,

L(f,P) $\displaystyle \leq $ L(f,R) $\displaystyle \leq $ U(f,R) $\displaystyle \leq $ U(f,Q)

This deduces that L(f,P) $\displaystyle \leq $ U(f,Q).

Therefore,

sup L(f,P) $\displaystyle \leq $ inf U(f,P).... which is the same as

lower riemann integral < upper riemann integral..

Is this proof totally correct? or is there another solution to proving this? thanks a lot :) xxx