Consider the continuous linear functionals $\ell_n, \; n\geq 0$ defined on $C([0,1])$ by $ \ell_0(f) = \int_0^1 f(t)dt, \;\; \ell_n(f) = \frac{1}{n} \sum_{k = 0}^{n-1} f(\frac{k}{n}), n\geq 1, \;\; f\in C([0,1])$
Show that $(\ell_n)$ converges weak* to $\ell_0$. Basic Riemann integration can be used without proof.
I have not read any measure theory, can someone please give me an simple proof for this? for me it seems "obvious" that the $\lim_{n\rightarrow \infty}|\ell_0(f) - \ell_n(f)| \rightarrow 0$. But what theorems should be used? Indicator functions and some kind of point wise approximation by constant functions? I've seen similar techniques before.