Existence of Limit iff $x',x'' > X, |f(x')-f(x'')| < \epsilon$

Question

I was given a theorem in class regarding uniform continuity that does not appear in my textbook. It says that

$$\lim_{x \to \infty} f(x) = a \iff \text{ for all } x',x'' > X, |f(x')-f(x'')| < \epsilon$$

Can anyone help me understand this theorem (not the proof, as i can follow the proof), or reference a website from which I can learn about context of this theorem?

Answer 1

This is the same, because the definition on infinite is, there exist some $M >0$ such that if $x,y > M$ then $|f(x) - a| < \varepsilon/2$ and $|f(y) - a| < \varepsilon/2$ hence$$|f(x) - f(y)| = |f(x) - a + a - f(y)| \leq |f(x) - a| + |a - f(y)| < \varepsilon/2 + \varepsilon/2$$