How can I formally write $f(x) \to 0$ when $x \to \infty$

Question

I've just proven that if $f:\mathbb{R} \to \mathbb{R}$ is uniformly continuous in $[a,b]$ and it is also uniformly continuous in $[b,+\infty)$ then $f$ is uniformly continuous in $\mathbb{R}_{\geq a}$. Now I am dealing with an exercise that says, $f:\mathbb{R} \to \mathbb{R}$ continuous and $f \to 0$ when $x \to \pm \infty$, I have to show that $f$ is uniformly continuous in $\mathbb{R}$.

I can picture the solution by defining an interval $[a,b]$ with $a<0$ and $b>0$, and using that $\left.f\right|_{[a,b]}$ is uniformly continuous, and together with the previous exercise, I can complete the proof of this one. However, I realized I don't know how to formally write that $f \to 0$ when $x\to \infty$.

Answer 1

To directly answer your question: formally, we can write that $\lim_{x \to \infty}f(x) = 0$ means that

For any $\epsilon > 0$, there exists an $r > 0$ such that $|f(x)| < \epsilon$ whenever $x > r$.