General Cesaro summation with weight

Assume that $a_n\to \ell $ is a convergent sequence of complex numbers and $\{\lambda_n\}$ is a sequence of positive real numbers such that $\sum\limits_{k=0}^{\infty}\lambda_k = \infty$

Then, show that, $$\lim_{n\to\infty} \frac{1}{\sum_\limits{k=0}^{n}\lambda_k} \sum_\limits{k=0}^{n}\lambda_k a_k=\ell =\lim_{n\to\infty} a_n$$

(Note that : This is more general than the special case where, $\lambda_n= 1$)

First you can assume that the $(a_n)$ are real, by considering the sequences $(\operatorname{Re} a_n)_n$ and $(\operatorname{Im} a_n)_n$ separately.

And then it is an immediate application of the Stolz–Cesàro theorem to $$ A_n := \sum_{k=0}^n \lambda_n a_n \quad, \quad B_n := \sum_{k=0}^n \lambda_n $$ since $(B_n)$ is strictly increasing and unbounded, and $$ \frac{A_{n+1} - A_n}{B_{n+1} - B_n} = a_{n+1} \to l $$

Let $\varepsilon >0$ and $N$such that $|a_k-l|\le \varepsilon $ for all $k>N$ Then, for $n>N$ we have, \begin{split}\left| \frac{\sum_\limits{k=0}^{n}\lambda_k a_k}{\sum_\limits{k=0}^{n}\lambda_k} -l\right| &= &\left| \frac{\sum_\limits{k=0}^{n}\lambda_k (a_k - l)}{\sum_\limits{k=0}^{n}\lambda_k} \right|\\ &= &\left| \frac{\sum_\limits{k=0}^{N}\lambda_k (a_k - l)+\sum_\limits{k=N}^{n}\lambda_k (a_k - l)}{\sum_\limits{k=0}^{n}\lambda_k} \right|\\ &\le & \frac{M}{\sum_\limits{k=0}^{n}\lambda_k} + \frac{\sum_\limits{k=N}^{n}\lambda_k \underbrace{\left| a_k - l\right|}_{\le\varepsilon}}{\sum_\limits{k=0}^{n}\lambda_k} \\ &\le& \frac{M}{\sum_\limits{k=0}^{n}\lambda_k} + \varepsilon\to 0 \end{split} since $\sum_\limits{k=0}^{N}\lambda_k\to \infty$. Where $M= \left|\sum_\limits{k=0}^{N}\lambda_k( a_k-l)\right|$