$\lim_{n\to \infty} \sqrt n(A_{n+1}-A_n) $ when $ A_n = \frac{1}{n}(a_1+a_2+\cdots+a_n)$ and $|a_n| \leq 1$ Let $a_n$, $n \geq 1$ be a sequence of real numbers satisfying $|a_n| \leq 1$ for all $n$. Define $$ A_n = \frac{1}{n}(a_1+a_2+\cdots+a_n)$$  for $n\geq 1$. Then $\lim_{n\to \infty} \sqrt n(A_{n+1}-A_n) $ is  A. $0$  B. $-1$  C. $1$  D. None of them  ATTEMPT  I tok sequence $a_n=1$ and answer came out to be 0. But how do i choose between A and D.  Thanks
Compute: $\lim_{n \rightarrow \infty}\sqrt{n}(A_{n+1} − A_n)$ where $A_n = \frac{1}{n}(a_1 + a_2 + \cdots + a_n)$ Let $\{a_n\}, {n\geq 1}$, be a sequence of real numbers satisfying $|a_n|\leq 1$ for all $n$. Define  $$A_n = \frac{1}{n}(a_1 + a_2 + \cdots + a_n),$$ for $n\geq 1$. Then find $\displaystyle\lim_{n \rightarrow \infty}\sqrt{n}(A_{n+1} − A_n)$ .    I proceed in this way  $$\lim_{n \rightarrow \infty}\sqrt{n}(A_{n+1} − A_n)=\lim_{n \rightarrow \infty}\sqrt{n}\left[\frac{1}{n+1}(a_1 + a_2 + \cdots + a_n+a_{n+1})-\frac{1}{n}(a_1 + a_2 + \cdots + a_{n})\right]=\lim_{n \rightarrow \infty}\left[{(na_{n+1}-a_1 - a_2 - \cdots - a_n})\frac{1}{\sqrt{n}(n+1)}\right]$$  Please help me to complete from here
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