Cauchy criteria at infinity

Question

i couldn't find the definition for Cauchy for functions as $x$ approaches infinity. I wrote it myself and i wanted to know if someone can correct me.

Let $f(x)$ be a function defined on all the real numbers. for every $ε>0$ there is an $M$ such that if $x,y>M$ so $|f(x)-f(y)|<ε$

thank you for your help!

Answer 1

If your goal is to define a criterion similar to the Cauchy criterion for sequences (that is, a sequence converges if and only if it is a Cauchy sequence), then you're right: $\lim_{x\to+\infty}f(x)$ exists (in $\mathbb R$) if and only if $f$ satisfies your condition.