# Establishing a simple upper bound (with one less parameter) to an unwieldy function with non-negative discrete parameters

I have this function at hand with 3 non-negative integer parameters that I would like to put a bound to:

$$f(n, b, d) = 1 - \left(1 - \left(\frac1n\right)^{b+d-1}\right)^{\left(n^{b-1}\right)}$$

My end goal is to show that for any positive $$\varepsilon_0$$ there is a $$n_0$$ and $$d^*$$ such that for all $$n \ge n_0$$ and $$b > 0$$, $$f(n, b, d^*) < \varepsilon_0$$. In simpler terms, I want to show that $$f$$ can be made arbitrarily small for any $$n$$ larger than a constant by affixing $$d$$ to be another large-enough constant.

To do this, I am looking for a decreasing function $$g(n, d)$$ which satisfies $$f(n, b, d) \le g(n, d)$$ for all $$b$$. Playing around with $$f$$ in Desmos's graphing calculator, it seems that $$g(n, d) = \left(\frac{2}{d}\right)^d$$ is such a function and seems very tight. link to graph

I don't need it to be tight. I can make do with any other decreasing function that allows me to reach my goal. (I can even do away with the bounding function altogether so long as I can reach my goal, but I still am curious about how one can bound a function like $$f$$.)