A cartoon of the Stolz–Cesàro Theorem // La caricatura di Teorema Stolz-Cesàro

 In the Mathematical Review of Students from Timișoara 1/1978, RMT for short, I found a problem (page 69) that I solved in my time :

 "3304.  Given the sequences  $\{a_n\}_{n\in\mathbb{N}}\;,\;\;\{b_n\}_{n\in\mathbb{N}}$ with the properties :

 (i)  $\{b_n\}_{n\in\mathbb{N}}$  is strictly monotone and unbounded.

                               (ii)  exists  $\displaystyle \lim_{n\to \infty}\frac{a_n}{b_n}$

                               (iii)  $\frac{a_{n+1}}{a_n}+\frac{b_{n+1}}{b_n}=2\;,\;\;(\forall) n\in\mathbb{N}.$

                           Prove that  $\displaystyle \lim_{n \to \infty}\frac{a_n}{b_n}=0.$

{author :} Titu ANDREESCU, student, Timișoara"

Solution CiP

              The condition  (iii)  is written equivalently : $\frac{a_{n+1}}{a_n}-1+\frac{b_{n+1}}{b_n}-1=0\;\Leftrightarrow$

$\Leftrightarrow\;\frac{a_{n+1}-a_n}{a_n}=-\frac{b_{n+1}-b_n}{b_n}\;\Leftrightarrow\;\frac{a_{n+1}-a_n}{b_{n+1}-b_n}=-\frac{a_n}{b_n} \tag{1}$

By  (ii)  the limit  $\displaystyle\lim_{n\to \infty}\frac{a_n}{b_n}$  exists, and then from  (1)  it follows that the limit  $\displaystyle \lim_{n\to\infty}\frac{a_{n+1}-a_n}{b_{n+1}-b_n}$  also exists. Then, according to Stolz–Cesàro theorem

$\displaystyle \lim_{n\to \infty}\frac{a_n}{b_n}=\lim_{n\to\infty}\frac{a_{n+1}-a_n}{b_{n+1}-b_n}\overset{(1)}{=}-\lim_{n\to\infty}\frac{a_n}{b_n}$,

hence  $\displaystyle \lim_{n\to \infty}\frac{a_n}{b_n}=0.$

QED

$\blacksquare$