Suppose $\ f: [0,1]$ -> $R$ is an increasing function. For each $ n \in N$ , consider the uniform partition $Pn := ( x_0, x_1,..., x_n )$ of $[0,1]$ where $x_i = i$ for $0≤i≤n$.
Now, I need to prove
a) $U(P_n,f)−L(P_n,f)$ = $ \frac{f(1)−f(0)}{n}$
and
b) show $f \in R\ [0, 1]$
I get some idea on part a: $m_i = \frac {x_i}{n} $ and $M_i= {x_i}{n}$ , then I am not sure
but for part b, I have no idea (is it something like $\epsilon > 0$ )?
Abridged solution. If $f$ is increasing on $[a, b]$ then $\sup\limits_{x \in [a, b]} f(x) = f(b)$ and $\inf\limits_{x \in [a, b]} f(x) = f(a).$ Q.E.D.
Addendum. Part (b) follows instantaneously from part (a) if you know the definition of Riemann integral as the equality if the supremum of lower approximations and infimum of upper approximations.