Convergence of a certain sequence defined almost like an average I am stuck on the following problem: 
Prove or disprove: If the sequence $a_n$ converges to $0$ as $n\to\infty$, then the sequence $$
\frac{a_{n+1}+a_{n+2}+ \cdots + a_{2n}}{\sqrt{n}}
$$
 is convergent.
The suggested edit is incorrect ...it is a sub n+1 etc.....

Consider $a_n = n ^{-1/4}$.
Then $a_n \to 0$ as $n \to \infty$, but since $a_k > (2n)^{-1/4}$ for $k = n+1, \ldots,2n-1 $ we have 
$$\frac1{\sqrt{n}}\sum_{k=n+1}^{2n}a_k > \frac{n}{(2n)^{1/4}n^{1/2}} = \frac{n^{1/4}}{2^{1/4}}\to \infty$$
