I have a function that maps from $\mathbb N$ to $\mathbb R$ and I need to show whether it is increasing.
Specifically, the function is $$f(n)=n \left[ \left(\alpha + \frac{1-\alpha}{n} \right)^{\delta} -\left(\frac{1-\alpha}{n} \right)^{\delta} \right],$$ where $n$ is a strictly positive integer, $\delta \in (1,2)$ and $\alpha \in (0,0.5)$ are real numbers (and parameters in my model). I want to show that $f(n+1) > f(n)$.
I can show that, if I treat $n$ as a real number and I take the derivative, it is always strictly positive for $n \geq 1$. Would this be enough to claim that the function defined over $\mathbb N$ is increasing as well?
Acording to theorem this should be enough, but the only hiccup I see is that with natural numbers we won't have an interval. In addition, there are other properties of this sequence that we are not aware that make it difficult to use the derivative.