I am trying to understand the proof of the theorem that if $f$ is continuous on a compact set $K \subseteq \mathbb{R}$ then $f$ is uniformly continuous on $K$. Here is the proof:
I am stuck on a couple things:
(1) How is $\lim [(y_{n_k})-(x_{n_k})] = 0$?
(2) The last statement of the proof claims that this proof has produced the desired contradiction. However, I don't understand how $\left| f(x_n) - f(y_n) \right| \geq \epsilon_0$ was contradicted by concluding that $\lim_{k \to \infty} \left| f(x_{n_k}) - f(y_{n_k}) \right| = 0$.
(3) [Edited from (2)] How does $\lim_{k \to \infty} \left| f(x_{n_k}) - f(y_{n_k}) \right| = 0$ imply $ \left| f(x_{n_k}) - f(y_{n_k}) \right| \geq \epsilon_0$ (in other words, where did the $\lim_{k \to \infty}$ part disappear)?
Any help is greatly appreciated!