The Map of Consecutive Differences of a Finite List of Reals Exists, and Its Sum Telescopes

In the setting of The Real Line: Standing Notation and Background for Calculus on Intervals, let $n\in\mathbb{N}$, let $[n]$ and $[n+1]$ be as in Intervals of Natural Numbers (ii), and let $d:[n+1]\to\mathbb{R}$ be a map. For $k\in[n]$ we have $k\in\mathbb{N}$ and $k\le n$ by Intervals of Natural Numbers: Initial Segments, Adding One Element, Splitting and Shifting (i), hence $k+1\in\mathbb{N}$ by Natural Numbers Are the Successors in Omega: One Is Least and Not a Successor of a Natural Number, the Successor Is Injective, and N Is Closed under Addition and Multiplication (vi) and $k+1\le n+1$ by Arithmetic and Order of the Natural Numbers (ix), so $k+1\in[n+1]$ by Intervals of Natural Numbers: Initial Segments, Adding One Element, Splitting and Shifting (i); and $1\in[n+1]$, as $n+1\in\mathbb{N}$ by Natural Numbers Are the Successors in Omega: One Is Least and Not a Successor of a Natural Number, the Successor Is Injective, and N Is Closed under Addition and Multiplication (vi) and $1\le n+1$ by Arithmetic and Order of the Natural Numbers (x), so that Intervals of Natural Numbers: Initial Segments, Adding One Element, Splitting and Shifting (i) applies.

(i) There is exactly one map $g:[n]\to\mathbb{R}$ with $g(k)=d(k+1)-d(k)$ for every $k\in[n]$, and its finite sum satisfies $\sum_{k=1}^{n}g(k)=d(n+1)-d(1)$.