how to prove $f(x)= x^2$ is uniformly continuous on $\mathbb{N}$

Question

Question is

$f(x)=x^2$ is uniformly continuous on $\mathbb{N}$

I know $f$ is uniformly continuous on bounded set because $\delta$ can be found easily from $|x^2-y^2|=|x-y||x+y|<||x|+|y|||x-y|<2C|x-y|$ for some positive number $C$

but if $Domf$ is $\mathbb{N}$, how can I find $\delta$?

Answer 1

For any $\epsilon$, let $\delta=1/2$. Then whenever $|x-y|<1/2$, then $x=y$, so $|x^2-y^2|<\epsilon$