nondecreasing f prove integrable with upper and lower sums

Question

I have been given a question as such:

If $f$ is nondecreasing on $[0,1]$ use upper and lower sums to prove $f$ is Riemann Integrable. So I know that using upper and lower sums to prove a riemann integral I have to show that the upper sum equals the lower sum and I know that

the upper sum = $\sum_{j=1}^{n} M_j(x_j-x_{j-1})$

the lower sum = $\sum_{j=1}^{n} m_j(x_j-x_{j-1})$

Where the x's are the size of the intervals and the big M is the upper height and little m is lower height. I'm just not so sure how I should go about using the fact f is non decreasing to prove the lower and upper sums equal each other.

Answer 1

If $f$ is non decreasing you can give explicit expressions in terms of $f$ for $M_i$ and $m_i$.

Answer 2

If $f$ is nondecreasing, then the lower sums would be strictly increasing and bounded above by the upper sums. Simply apply the monotone convergence theorem from there.