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!
Best Answer
First question: If a sequence tends to $0$ so does every subsequence. Since $y_n-x_n \to 0$ so does $y_{n_k}-x_{n_k}$.
Second question: If $|c_n| \geq \epsilon_0$ for all $n$ then $|c_{n_k}| \geq \epsilon_0$ for all $k$. This implies that $c_{n_k}$ cannot tend to $0$.