Summation approaches zero

Question

if I have a sequence $A=$ $\frac{1}{n} \sum _{k=1}^n kC_k$ and $nC_n$ approaches $0$ as $n$ approaches infinity, then how can I show that $A$ goes to $0$? I just need it for my proof of the Tauber's theorem.

Answer 1

Since $kC_k \to 0$, for any $\epsilon >0$ there exists $N \in \mathbb{N}$ such that for $k \geqslant N$ we have $kC_k > -\epsilon$.

Hence

$$\frac1{n}\sum_{k=1}^nkC_k = \frac1{n}\sum_{k=1}^N kC_k+ \frac1{n}\sum_{k=N+1}^n kC_k\geqslant \frac1{n}\sum_{k=1}^N kC_k - \frac{n-N}{n}\epsilon,$$

and for any $\epsilon > 0$,

$$\liminf_{n \to \infty} \frac1{n}\sum_{k=1}^nkC_k \geqslant -\epsilon \implies \liminf_{n \to \infty} \frac1{n}\sum_{k=1}^nkC_k = 0.$$

Make a similar argument for $\limsup$ and conclude.