Prove explanation for exponential growth

Question

I'm currently learning about Algorithms and Data Structures, and I have a proof in my lecture notes that I don't understand.

The Theorem: $n^d \in o((1+\epsilon)^n)$ with $\epsilon >0$ and $n,d \in \mathbb{N}$.

"The Proof": $\lim_{n \rightarrow \infty}\frac{n^d}{(1+\epsilon)^n} = 0$

I don't really know why this "statement" is enough to prove our theorem. I know about limits etc. from calculus but I don't know why we can use this knowledge here.

Answer 1

From the definition of little-O notation,

$$f \in o(\phi) \text{ provided that } f/\phi \to 0 \tag{*}\label1$$

In your "theorem", take $f: n \mapsto n^d$ and $\phi: n \mapsto (1+\epsilon)^n$ and apply \eqref{1}.$\tag*{$\square$}$