Suppose $\Sigma c_n $ is cesaro summable to $0$, prove that for every $0\le r <1$ , $\Sigma c_n r^n$ is convergent.
(A series is cesaro summable if the sequence of the means of its partial sums converges)
If we know this , I know how to prove that Abel sum tends to $0$ ,when $r $ tends to $1$, and then the proof of "cesaro summablity implies Abel summablity " is complete .
help me please. Thank