Given a real positive sequence $\{a_n\}$ such that $\sum_{i=1}^n a_i$ converges.
Prove that there exists a real sequence $\{c_n\}$ monotonically increasing to $\infty$ such that $\sum_{i=1}^\infty a_nc_n$ converges.
What I have done:
I have written $$\sum_{i=1}^{N+1}a_ic_i=c_{N+1}\sum_{i=1}^{N+1}a_i-\sum_{i=1}^N(c_{n+1}-c_n)S_n$$
and then try to prove the convergence by the Cauchy criterion. However my evaluation is still not good enough. I'm really stuck.
Here is an alternative approach. There is a sequence of positive integers $n_1<n_2<n_3<\cdots$ such that $\sum_{n=n_k}^\infty a_n < 2^{-k}$. Define $c_n = k$ when $n_k\leq n<n_{k+1}$ (with $n_0=1$). Then $c_n\nearrow+\infty$ and
$$\sum_n a_nc_n=\sum_{k=1}^\infty\sum_{n=n_k}^{n_{k+1}-1}a_nc_n\leq\sum_{k=1}^\infty k2^{-k}<\infty.$$
Put $\displaystyle R_n=\sum_{k\geq n} a_k$. Then $R_n>0$, and $R_n$ decrease to $0$ as $n\to +\infty$. We show that $\displaystyle c_n=\frac{1}{\sqrt{R_n}}$ (obviously increasing to $+\infty$) do the job.
We have: $$a_n c_n=\frac{a_n}{\sqrt{R_n}}=\frac{R_n-R_{n+1}}{\sqrt{R_n}}\leq \int_{R_{n+1}}^{R_n}\frac{dx}{\sqrt{x}}$$
Hence for $N \geq 1$:
$$\sum_{n=1}^N a_n c_n \leq \int_{R_{N+1}}^{R_1}\frac{dx}{\sqrt{x}}\leq \int_0^{R_1}\frac{dx}{\sqrt{x}}<+\infty$$ Hence $\sum a_n c_n$ is convergent.
