# Cesaro summability implies bounded partial sums

by stripes
Tags: bounded, cesaro, implies, partial, summability, sums
 P: 263 1. The problem statement, all variables and given/known data Suppose $c_{n} > 0$ for each $n\geq 0.$ Prove that if $\sum ^{\infty}_{n=0} c_{n}$ is Cesaro summable, then the partial sums $S_{N}$ are bounded. 2. Relevant equations -- 3. The attempt at a solution I tried contraposition; that was getting me nowhere. I have a few inequalities here and there but they don't tell me anything. I need to show that there exists an upperbound for the partial sums. This means there exists a least upperbound. I need to find that least upperbound. Because$c_{n} > 0$ for each $n\geq 0,$ then the series is nondecreasing, which means the partial sums are nondecreasing, so we are looking for an upperbound, not a lowerbound.
 Quote by stripes 1. The problem statement, all variables and given/known data Suppose $c_{n} > 0$ for each $n\geq 0.$ Prove that if $\sum ^{\infty}_{n=0} c_{n}$ is Cesaro summable, then the partial sums $S_{N}$ are bounded. 2. Relevant equations -- 3. The attempt at a solution I tried contraposition; that was getting me nowhere. I have a few inequalities here and there but they don't tell me anything. I need to show that there exists an upperbound for the partial sums. This means there exists a least upperbound. I need to find that least upperbound.
That least upper bound, if it exists, is $\lim_{n \to \infty} S_n$, which is the definition of $\sum_{n=0}^{\infty} c_n$ in the traditional sense.