Equivalence of Definitions of Total Variation of Real Function on Closed Bounded Interval

Proof
We aim to prove that:


 * $\ds \sup_{P \in \map X {\closedint a b} } \paren {\map {V_f} {P ; \closedint a b} } = \sup_{\mathcal S \in \map {\mathcal P_F} {\closedint a b} } \paren {\map {V_f^\ast} {\mathcal S ; \closedint a b} }$

We will prove that:


 * $\ds \sup_{P \in \map X {\closedint a b} } \paren {\map {V_f} {P ; \closedint a b} } \le \sup_{\mathcal S \in \map {\mathcal P_F} {\closedint a b} } \paren {\map {V_f^\ast} {\mathcal S ; \closedint a b} }$

and:


 * $\ds \sup_{\mathcal S \in \map {\mathcal P_F} {\closedint a b} } \paren {\map {V_f^\ast} {\mathcal S ; \closedint a b} } \le \sup_{P \in \map X {\closedint a b} } \paren {\map {V_f} {P ; \closedint a b} }$

Note that:


 * $\map X {\closedint a b} \subseteq \map {\mathcal P_F} {\closedint a b}$

Note also that if $\mathcal S$ is a finite subdivision of $\closedint a b$, we have:


 * $\map {V_f} {\mathcal S ; \closedint a b} = \map {V_f^\ast} {\mathcal S ; \closedint a b}$

So, we have:


 * $\set {\map {V_f} {P ; \closedint a b} : P \in \map X {\closedint a b} } \subseteq \set {\map {V_f^\ast} {\mathcal S ; \closedint a b} : \mathcal S \in \map {\mathcal P_F} {\closedint a b} }$

So, from Supremum of Subset we obtain:


 * $\ds \sup_{P \in \map X {\closedint a b} } \paren {\map {V_f} {P ; \closedint a b} } \le \sup_{\mathcal S \in \map {\mathcal P_F} {\closedint a b} } \paren {\map {V_f^\ast} {\mathcal S ; \closedint a b} }$

In Equivalence of Definitions of Bounded Variation for Real Function on Closed Bounded Interval, it is shown that for each $\mathcal S \in \map {\mathcal P_F} {\closedint a b}$, there exists a finite subdivision $\mathcal S^\ast \in \map X {\closedint a b}$ such that $\mathcal S \subseteq \mathcal S^\ast$ and:


 * $\map {V_f^\ast} {\mathcal S ; \closedint a b} \le \map {V_f} {\mathcal S^\ast ; \closedint a b}$

which establishes:


 * $\ds \sup_{\mathcal S \in \map {\mathcal P_F} {\closedint a b} } \paren {\map {V_f^\ast} {\mathcal S ; \closedint a b} } \le \sup_{P \in \map X {\closedint a b} } \paren {\map {V_f} {P ; \closedint a b} }$