Why can the chain rule be proven by showing that the set $M = \{ n \in \mathbb{Z}^+ : f(x_n) = f(a) \}$ is finite?

Question

I am currently trying to understand this proof of Proposition 26 in my lecture notes on the chain rule.

In these notes, it says that proposition 26 can be proven by showing that the set $M = \{ n \in \mathbb{Z}^+ : f(x_n) = f(a) \}$ is finite. Can anyone please explain to me why this is the case?

Answer 1

Suppose that the set $M$ is infinite. Then there is a subsequence $(x_{n_k})$ of $(x_n)$ with

$f(x_{n_k})=f(a)$ for all $k$. Hence $f(x_{n_k})-f(a)=0$ for all $k$. Therefore

$f'(a) = \lim_{k \to \infty}\frac{f(x_{n_k})-f(a)}{x_{n_k}-a}=0$, a contradiction, since in Proposition 26 we have $f'(a) \ne 0$.