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

Theorem
Let $a, b$ be real numbers with $a < b$.

Let $f : \closedint a b \to \R$ be a real function.

Definition 1 implies Definition 2
Suppose that there exists a $M \in \R$ such that:


 * $\map {V_f} {P ; \closedint a b} \le M$

for all finite subdivisions $P$.

Let $\mathcal S$ be a finite non-empty subset of $\closedint a b$.

Define:


 * $\mathcal S^\ast = \mathcal S \cup \set {a, b}$

and write:


 * $\mathcal S^\ast = \set {x_0, x_1, \ldots, x_n}$

with:


 * $a = x_0 < x_1 < x_2 < \cdots < x_{n - 1} < x_n = b$

Then $\mathcal S^\ast$ is a finite subdivision of $\closedint a b$ and:

so:


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

for all finite non-empty subset of $\closedint a b$.

Definition 2 implies Definition 1
Suppose that there exists a $M \in \R$ such that:


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

for all finite non-empty subsets $\mathcal S$ of $\closedint a b$.

Note that any finite subdivision of $\closedint a b$ is also a finite non-empty subset of $\closedint a b$.

So, in particular, for any finite subdivision $P$ of $\closedint a b$, we have:


 * $\map {V_f} {P ; \closedint a b} = \map {V_f^\ast} {P ; \closedint a b} \le M$

as required.