How can we prove that [tex]n^s-(n-1)^s[/tex] converge to zero as [tex]n \to \infty[/tex] where s as a real number satisfies [tex]0<s<1[/tex]? I am specifically looking for a more or less elementary proof for this for real s. I think we can use the infinite binomial expansion, but I am looking for something that does not require more than elementary calculus.
I thought about the graph of that function and why the limit is "obvious" from the graph. Translating the graph picture into mathematics, I think an easy way is the mean value theorem. With it you find [tex] n^s-(n-1)^s=s(n+\xi)^{s-1}\to 0 [/tex]