**Lemma** For $\alpha \in \mathbb{R}$ with $\alpha>0$, $n^{-\alpha} \to 0$ as $n \to \infty$. **Proof** Given $\epsilon>0$, By [[Archimedean Property of Real Numbers|AP]] $\exists N \in \mathbb{N}$ such that $N>\epsilon^{-\frac{1}{a}}$. Using [[Inequalities for powers of positive reals with natural number exponents]], then for all $n \geq N$, $0 \leq n^{-\alpha}\leq N^{-\alpha}>\epsilon$ so $|n^{-\alpha}|<\epsilon \quad \square$