Let$\{x_n\}$ be a sequence such that there exists a $0<C<1$ such that $$|x_{n+1} - x_n | \le C|x_n - x_{n-1}|.$$ Prove that $\{x_n\}$ is Cauchy. Hint: You can freely use the formula (for $C \not=1$) $$1+C+C^2+...+ C^n = \frac{1-C^{n+1}}{1-C}.$$
Let $x_n - x_{n-1} = y_n$. Then, $\frac{|y_{n+1}|}{|y_n|}\le C < 1$. This implies that $y_n$ converges to $0$ by ratio test. I also have the theorem that $y_n$ converges iff $y_n$ is Cauchy. But, I think that this does not mean $x_n$ is Cauchy. Could you give me some help? How can I use the hint given?