# Notation in Cauchy Sequence and Convergence

I have been reading lecture script and wasn't sure where this inequation for a fix-point iteration comes from and what it means.

For $$k, j \geq 0$$ $$\left|x_{k+j}-x_{k}\right| \leq \sum_{i=0}^{j-1}\left|x_{k+i+1}-x_{k+i}\right| \leq\left|x_{k+1}-x_{k}\right| \sum_{i=0}^{j-1} q^{i} \leq \frac{q^{k}}{1-q} | x_{1}-x_{0}|$$

a. When does the above inequation turn into an equation ? b. How do we come to the inequations betwteen the 4 terms c. I have a vague idea that the last term in the equation has a connection with the a-posteri error estimation but i am not sure what it has to do with the Cauchy sequence

• The first inequality is because $$x_{k+j}-x_k = \sum\limits_{i=0}^{j-1}\left(x_{k+i+1} - x_{k+i}\right),$$ and then use the triangle inequality. – Minus One-Twelfth Apr 13 at 13:02