Riemann sum of Lipschitz function

Question

Let $f : [0,1] →\mathbb{R}$ be a Lipschitz function such that $|f(x) − f(y)| ≤ λ|x − y|$ for all $x,y ∈ [0,1]$. Let ̇$P$ be a tagged partition of $[0,1]$ such that $|| ̇P||< \frac{1}{m}$ for some $m ∈\mathbb{N}$. Show that $| \int_0^1f(x)dx - S(f,P)| < \frac{λ}{m}$.

I know I have to use the upper and lower Riemann sums here and somehow relate it to the Lipschitz definition. I have no clue where to start for that, though.

Answer 1

Hint:

Let $0=x_0<x_1\dots<x_n=1$ be a partition of $[0,1]$ with $\|P\|<\frac{1}{M}$. Write \begin{align} \left|\int_0^1 f-S(f,P) \right|&=\left|\sum_{k=0}^{n-1}\int_{x_k}^{x_{k+1}} f-S(f,P) \right| \\ &\stackrel{MVT}{=}\left|\sum_{k=0}^{n-1}f(x_k^*)(x_{k+1}-x_k)-S(f,P) \right| \end{align}