Multivariable Calculus proof explanation help, Cauchy Sequences

Question

Proof here. This is a proof from the Advanced Calculus book by Gerald.B Folland. I understand all the steps except where the author goes on to say that $|\textbf{x}_k|<|\textbf{x}_{K+1}|+1$ for all $k>K$. How did he get to this inequality?

Answer 1

It's just using the definition of Cauchy convergence with $\epsilon=1$.

For all $k,k'> K$ you have $|x_k-x_{k'}|< \epsilon = 1$. In particular for $k' = K+1$ you have that for all $k> K$

$$|x_k-x_{K+1}|< 1\,.$$

Since $|a|-|b|\leq |a-b|$ for all $a,b$, in particular you have $$|x_k|-|x_{K+1}|\leq |x_k-x_{K+1}|< 1\,,$$ from which the statement follows.