An algebra problem by Kushal Bose

$\begin{aligned} A_n & = \frac 1n \left(a_1+a_2+a_3+\cdots + a_n \right) \\ A_{n+1} & = \frac 1{n+1} \left(a_1+a_2+a_3+\cdots + a_n + a_{n+1} \right) \\ & = \frac 1{n+1}\left(nA_n + a_{n+1} \right) \\ A_{n+1} - A_n & = \frac {nA_n}{n+1} + \frac {a_{n+1}}{n+1} - A_n \\ & = \frac {a_{n+1}-A_n}{n+1} \end{aligned}$

$\begin{aligned} \implies L & = \lim_{n \to \infty} \sqrt n \left(A_{n+1}-A_n\right) \\ & = \lim_{n \to \infty} \frac {\sqrt n \left(a_{n+1}-A_n\right)}{n+1} & \small \color{#3D99F6} \text{Dividing up and down by }\sqrt n \\ & = \lim_{n \to \infty} \frac {a_{n+1}-A_n}{\sqrt n+ \frac 1{\sqrt n}} & \small \color{#3D99F6} \text{Since }|a_n| < 1 \ \forall n, |A_n| < 1 \\ & = \boxed{0} \end{aligned}$