# Proof of Telescoping Series

I am trying to prove the properties of the telescoping series via an exercise in Tao's analysis text. The exercise, with the full proposition filled in, is:

Let $$(a_n)_{n=0}^{\infty}$$ be a sequence of real numbers which converge to $$0$$, i.e., $$\lim\limits_{n \to \infty} a_n = 0$$. Then the series $$\sum\limits_{n=0}^{\infty} (a_n - a_{n+1})$$ converges to $$a_0$$. How does the proposition change if we assume that $$a_n$$ does not converge to zero, but instead converges to some other real number, $$L$$?

Here is my attempt at a proof.

Per the hint in Tao, we guess a formula for the $$n$$th partial sum, $$S_n$$, and prove it by induction.

Lemma: For all $$n \in \mathbb{N}$$, $$S_n = a_0 - a_{n+1}$$.

Base Case: $$S_0 = \sum\limits_{n=0}^{0} (a_n - a_{n+1}) = a_0 -a_1$$.

Induction Hypothesis: Assume $$S_k = \sum\limits_{n=0}^{k} = a_0 - a_{k+1}$$.

Induction Step: We prove the stateent for $$n = k + 1$$. \begin{align*} s_{k+1} & =\sum\limits_{n=0}^{k+1} (a_n - a_{n+1}) \\ & = \sum\limits_{n=0}^k (a_n - a_{n+1}) + (a_ {k+1} - a_{k+2}) \\ & = s_k + (a_{k+1} - a_{k+2}) \\ & = a_0 - a_{k+1} + a_{k+2} - a_{k+2} \\ & = a_0 - a_{k+2} \end{align*} This closes the induction.

We have: \begin{align*} \lim s_n =\lim\limits_{n \to \infty} [a_0 - a_{n+2}] = \lim\limits_{n \to \infty} a_0 - \lim\limits_{n \to \infty} a_{n+2} = a_0 - 0 = a_0. \end{align*} And since the series converges to the same limit as the sequence of partial sums, we conclude that the series also converges to $$a_0$$.

If $$a_n$$ converges to $$L$$, then $$\lim\limits_{n \to \infty} a_{n+2} = L$$, so the sequence of partial sums would converge to $$a_0 - L$$, and thus the series would converge to $$a_0 - L$$.

If $$(b_n)_{n=0}^\infty$$ is a sequence of real numbers which converge to $$L$$ then we can just let $$a_n=b_n-L$$ such that $$a_n$$ converges to $$0$$ hence $$\sum_{n=0}^\infty (b_n-b_{n+1})=\sum_{n=0}^\infty (a_n+L-(a_{n+1}+L))=\sum_{n=0}^\infty (a_n-a_{n+1})=a_0=b_0-L$$