ThmDex – An index of mathematical definitions, results, and conjectures.
Result R3148 on D4675: Real series
Sum of real telescoping series

Let $x : \mathbb{N} \to \mathbb{R}$ be a D4685: Real sequence.
Then
 (1) $$N \in \{ 1, 2, 3, \ldots \} \quad \implies \quad \sum_{n = 0}^N (x_{n - 1} - x_n) = x_N - x_0$$ (2) $$\lim_{n \to \infty} x_n = 0 \quad \implies \quad \sum_{n = 0}^{\infty} (x_{n + 1} - x_n) = - x_0$$
Proofs
Proof 0
Let $x : \mathbb{N} \to \mathbb{R}$ be a D4685: Real sequence.
Proceeding by induction, the base case of $N : = 1$ is vacuously true: $$\sum_{n = 0}^N (x_{n + 1} - x_n) = (x_1 - x_0) = x_1 - x_0$$ Assume then that the equality $\sum_{n = 0}^N (x_{n + 1} - x_n) = x_N - x_0$ holds for some $N \geq 1$. Then $$\begin{split} \sum_{n = 0}^{N + 1} (x_{n + 1} - x_n) & = (x_{N + 1} - x_N) + \sum_{n = 0}^N (x_{n + 1} - x_n) \\ & = (x_{N + 1} - x_N) + x_N - x_0 \\ & = x_{N + 1} - x_0 \end{split}$$ The first claim therefore follows from R800: Proof by principle of weak mathematical induction. For the second claim, suppose that $x$ converges to $0$ as $n$ increases without bound. Taking limits on both sides of $\sum_{n = 0}^N (x_{n + 1} - x_n) = x_N - x_0$ as $N \to \infty$, one has $$\lim_{N \to \infty} \sum_{n = 0}^N (x_{n + 1} - x_n) = \lim_{N \to \infty} (x_N - x_0) = - x_0$$ $\square$