Originally Posted by

**paupsers** Yes, you're right, part of the hypothesis should be that the function is Riemann integrable.

Does this proof work?

Proof: We know $\displaystyle L(P,f)\leq S(P,f)\leq U(P,f)$

We also know that $\displaystyle L(P,f)\leq \int_a^bf\leq U(P,f)$

This implies that $\displaystyle L(P,f)\leq |S(P,f)-\int_a^bf|\leq U(P,f)$

Since f is integrable on [a,b], we know there exists a partition P such that $\displaystyle U(P,f)-L(P,f)<\epsilon$.

This implies that $\displaystyle |S(P,f)-\int_a^bf|<\epsilon$. QED.

Any advice?