Proof of Cauchy Criterion

Question

How did they get $|x_{n} - x_{n_{k}}| = \dfrac{\epsilon}{2}$ in the proof?

Answer 1

You know that $|x_n - x_m| < \epsilon/2$ for $n,m \geq N$. In the proof, $n$ and $n_K$ are greater than or equal to $N$. They don't mention that $n \geq N$, but that is what they have used.