Q:

Prove using the Sandwich Theorem for sequences

CLAIM: Prove with using the Sandwich Theorem that if $a_n$ convergent and $b_n→∞$
 then $\frac{a_n}{b_n}→0$
DO NOT USE: (Bounded $X$ $d_n→0$) $→0$

A:

Since ${a_n}$, there is a real constant number $A>0$ such that
$$
|a_n| \le A \quad \forall n.
$$
Also, because $\lim b_n=\infty$, there is an integer $N$ such that
$$
b_n>0 \quad \forall n\ge N.
$$
Hence, for all $n\ge N$ we have
$$
-\frac{A}{b_n} \le \frac{a_n}{b_n} \le \frac{A}{b_n},
$$
and therefore by the Sandwich Theorem we have
$$
\lim_{n\to \infty}\frac{a_n}{b_n}=0.
$$

