Let \(\left(\Pi_{n}ight)_{n \geqslant 1}\) be a sequence of refining (i.e. \(\left.\Pi_{n} \subset \Pi_{n+1}ight)\) partitions of \([0,1]\) such that \(\lim _{n ightarrow \infty}\left|\Pi_{n}ight|=0\). Show that for every function \(f \in \mathcal{C}[0,1]\) the limit

\[\begin{aligned}\lim _{n ightarrow \infty} S_{1}^{\Pi_{n}}(f ;[0,1]) & =\operatorname{VAR}_{1}(f ;[0,1]) \\& :=\sup \left\{S_{1}^{\Pi}(f ;[0,1]): \Pi \text { finite partition of }[0,1]ight\}\end{aligned}\]exists in \([0, \infty]\) (and is, in particular, independent of the sequence \(\left(\Pi_{n}ight)_{n \geqslant 1}\) ).