= Linear-growth bound for iterated Cesàro means
{title2=$|a_n|\leq Cn\Longrightarrow|C^k a_n|\leq Cn$}
Let $Ca_n=n^{-1}\sum_{j=1}^n a_j$ be the <Cesàro mean> of a <sequence>. If $|a_n|\leq Cn$ for every positive <integer> $n$, the <triangle inequality> gives $|Ca_n|\leq C(n+1)/2\leq Cn$. <Mathematical induction> repeats this estimate with the same constant for every iterate. An eventual $O(n)$ bound can be made global by increasing $C$ to absorb the finitely many initial terms. Consequently iterated <Cesàro means> preserve linear growth in <Big O notation>; this bound alone does not assert convergence or summability.
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