Let $D$ be any finite collection interval on $E$ and function $f:E\to\mathbb{R}$. If $E$ is a compact set, show that $$V(f,E)=\sup\{V(f,D) \mid \text{for } D \text{ is any finite collection interval on } E\}$$
My Attempt:
My ideas, we can prove it by two condition where $V(f,E)\geq \sup V(f,D)$ and $V(f,E) \leq V(f,D)$. Its easy to prove that $V(f,E)\geq \sup V(f,D)$ because $E$ is a compact set, so we have finite collection open cover on $E$ such that $$ E \subseteq \bigcup_{k=1}^{n}I_{k}.$$ Hence we have for any collection interval $D$ on $E$ we have \begin{align*}
V(f,E)\geq \sup\{V(f,D)\mid \text{for } D \text{ is any finite collection interval on }E\}
\end{align*}
How we prove $V(f,E) \leq \sup\{V(f,D) \mid \text{for } D \text{ is any finite collection interval on }E\} $? I don't have any idea. Thank you in advance