$\sum_{n = 1}^{\infty}\frac{x_{n}}{a_{n}}$ converges. Show that $\frac{1}{a_{n}}\sum_{k = 1}^{n}x_{k}\rightarrow 0$ as $n\rightarrow \infty$. Let $\{x_{n}\}_{n\geq 1}$ be a sequence in $\mathbf{R}$ and $\{a_{n}\}_{n\geq 1}$ be a sequence of positive real numbers satisfying $a_{n} \uparrow \infty$ as $n \rightarrow \infty$ . Further, suppose that $\sum_{n = 1}^{\infty}\frac{x_{n}}{a_{n}}$ converges. Then show that $\frac{1}{a_{n}}\sum_{k = 1}^{n}x_{k}\rightarrow 0$ as $n\rightarrow \infty$.
My Attempt: If $(x_{n})$ sequence of positive real number then by Comparision test $\frac{1}{a_{n}}\sum_{k = 1}^{n}x_{k}$ is converges to zero. But  How can I approach for arbitrary sequence $(x_{n})$.

Let $y_j$ be a sequence of $n$ real numbers.
If $|y_1+\ldots+y_n|\geq1$ and if $b_1\geq b_2\geq\ldots\geq b_n=1$,
then there exist $f$ and $g$ such that $\displaystyle\bigg|\sum_{j=f}^g b_j y_j\bigg|\geq1$.
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This is, I think, the key fact. If you want me to include the proof, let me know.
