Hi I was given the following problem. Let $a_{n}>0$ increasing monotonically to $\infty$ as $ n\to\infty$ and $\sum_{n=1}^{\infty}\frac{y_{n}}{a_{n}}$ is convergent. Use summation by parts to prove that $\lim_{n\to\infty}\frac{1}{a_{n}}\sum_{i=1}^{n}y_{i}=0$
My approach was let $Y_{n} = \sum_{i=1}^{n}y_{i}$ $$ \sum_{k=q}^{p} \frac{y_{k}}{a_{k}} = \sum_{k=q}^{p}Y_{k}(\frac{1}{a_{k}}-\frac{1}{a_{k+1}}) + \frac{Y_{p}}{a_{p+1}}-\frac{Y_{q-1}}{a_{q}} $$
since the sum of $\frac{y_{k}}{a_{k}} $ converges, the RHS can be bounded from above by $\epsilon > 0$ for q,p large enough as the partial sums $\sum^{n}_{k=1}\frac{y_{k}}{a_{k}} $ form a Cacuhy sequence. But how does one reach the conclusion that $\frac{Y_{k}}{a_{k}} \to 0$ or is the approach wrong?