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'Amortized' series implies limit of arithmetical mean = 0

Source: 38th Brazilian Undergrad MO (2016) - First Day, Problem 1

November 25, 2016
real analysis

Problem Statement

Let (an)n≥1(a_n)_{n \geq 1} s sequence of reals such that ∑n≥1ann\sum_{n \geq 1}{\frac{a_n}{n}} converges. Show that
lim⁡n→∞1n⋅∑k=1nak=0\lim_{n \rightarrow \infty}{\frac{1}{n} \cdot \sum_{k=1}^{n}{a_k}} = 0
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