Class K infinity function

Question

A continuous function $\alpha:[0,\infty)\to [0,\infty)$, is said to belong to class $\mathcal{K}$ if it is strictly increasing $\alpha(0) = 0$ and $\alpha(t) \to \infty$ as $t\to\infty$.

Let $s,r$ be some positive values, and $s >> r$. Is the following true?

$\lim_{s\to\infty} \alpha(s) - \alpha(s-r) \neq 0$ for any class $\mathcal{K}$ function $\alpha$.

Answer 1

No, you can take $f(u)=\ln (1+u).$