I'm kind of stuck on a question that I'm not sure how to approach.
Assume that $f(x) \rightarrow a$ for $x \rightarrow \infty$ and $f(x) \rightarrow b$ for $x \rightarrow -\infty$. Show that for every $\epsilon>0$ there exists an $M>0$ such that $|f(x)−f(y)|<\epsilon$ as long as $x,y>M$.
I'm not really sure how to approach this because I haven't seen the $|f(x)−f(y)|$ notation before and at the same time there's an "a" but also a "b" which makes it a bit different from what I've seen before. My thoughts so far is that I maybe could use one of the definitions from my book saying that: $f(x)\rightarrow a$ for $x \rightarrow \infty$ if for every $\epsilon>0$ there exists an $M>0$: $|| f(x)-a||<\epsilon$ for all $x>M$. But I'm also not sure about how to extract what I want from this. Any help is appreciated really!
Hint: $$|f(x)-f(y)|=|f(x)-a+a-f(y)|\le |f(x)-a|+|f(y)-a|$$
The "b" is a red herring, as $x,y$ are both large and positive.