Help please
Let $(a_{n})_{n>o}$ and $(b_n)_{n>o}$ be sequences of complex numbers and suppose that:
(i) The sequence of partial sums $S_{m}=\sum_{n=0}^{m}a_{n}$ is bounded
(ii) $\mathrm{lim}b_{n}=0$
(iii) The sum $\sum_{n=1}^{\infty} |b_{n} - b_{n-1}|$ is finite.
Prove that the series $\sum a_{n}b_{n}$ is convergent. Hint: Use "Abel summation":
$\sum_{k=n}^{m}a_{k}b_{k}=\sum_{k=n}^{m}(S_{k} - S_{k-1})b_{k}$ for $n>0$