Proof with Darboux integral - question.

[peripatein]

Hello,

Homework Statement

I was asked to prove that for f(x), Riemann integrable and bounded from above by real number M:
∫[a,b] f(x) < M(b-a)

Homework Equations

The Attempt at a Solution

Since f(x) is Riemann integrable, ∫[a,b] f(x) must be equal to both the lower and upper Darboux sums. Therefore: ∫[a,b] f(x) = supf(x) _[a,b]Ʃ[i=1,n]Δx_i = supf(x) _[a,b](b-a) <= M(b-a)
I am really not sure this is rigorous enough, or even how it ought to be approached. I'd appreciate some guidance please.

 

[jfgobin]

I think this can be seen as a consequence of the integral mean value theorem.