**Lemma** If $a_{n} \geq b_{n}$ and $b_{n} \to \infty$ as $n \to \infty$, then $a_{n} \to \infty$ as $n \to \infty$. Similarly [[Limits of Real Sequence Preserve Weak Inequalities]] & [[Sandwich Rule]]. **Proof** If $b_{n} \to \infty$ then $\forall R \in \mathbb{R}$, there exists $N$ such that $b_{n} > R$ for every $n \geq N$. Since $a_{n} \geq b_{n}$, we have $a_{n}>R$ for all $n \geq N$ so $a_{n} \to \infty$.