Is every single-valued real function on the set of all positive integers uniformly continuous

Question

According to the classroom notes "Uniformly Continuous Linear Set" in American Mathematical Monthly, Vol. 62. No. 8(Oct., 1955) pp. 579-580, Author: Norman Levine, DOI: 10.2307/2307254..

How to verify that every single-valued real function on the set of all positive integers is uniformly continuous??

Answer 1

Fix any $\delta < 1$. Then for any $\varepsilon > 0$ and $x, y \in \mathbb{N}$ with $|x-y| < \delta$ (which, in this case, means x = y) you have $$|f(x) - f(y)| = 0 < \varepsilon.$$