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Let $\sum a_n$ be a convergent series of positive terms ; then we know $\lim \inf (na_n)=0$ ; can we derive from here that if $\{a_n\}$ is decreasing , then $\lim (na_n)=0$ ?

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Hint:

For any $\epsilon > 0$ there exists $N \in \mathbb{N}$ such that $\displaystyle\epsilon > \sum_{k=n+1}^{2n}a_k> na_{2n}$ for all $n > N$

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