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Proposition. Let $f : \mathbb R \to \mathbb R$ be continuous and such that $\lim_{|x|\to+\infty} f(x) = 0$. Then $f$ is uniformly continuous.

How to prove such a statement?

The hypothesis means that for all $\varepsilon > 0$ I can find an $M > 0$ such that for all $x \in \mathbb R$ such that $|x| > M$ we have $|f(x)| \leq \varepsilon$. If I pick $\varepsilon > 0$ and find such $M$. Then by Heine's theorem $f$ is uniformly continuous on $[-M,+M]$. How do I establish uniform continuity on the rest of the domain?

giobrach
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  • https://math.stackexchange.com/q/1106275/219176 – Red shoes Jul 06 '17 at 18:37
  • the proof is almost finished, just use the definition of uniform cont. – Felice Iandoli Jul 06 '17 at 18:37
  • Won't the $\delta$ given by the uniform continuity of $f$ on $[-M,+M]$ also work for pairs of points taken outside of this closed interval, but both on the same side? It seemed too easy to close it off like that, so I asked. – giobrach Jul 06 '17 at 18:42

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