This document presents a part of a problem involving the Cesàro mean of a sequence and the Cesàro theorem in a general case. It describes a sequence $(u_n)$ that converges to a real number $\ell$, and it defines the Cesàro mean sequence $(v_n)$ based on it. The goal is to show certain properties and derive conclusions about the behavior of the Cesàro mean sequence.

Let me break down the tasks for each part:

1. We are given that the sequence $(u_n)$ converges to $\ell$, and we need to show the following:

(a) We need to show that there exists an integer $n_1 \in \mathbb{N}^*$ such that for all $n \geq n_1$, the absolute difference $|u_n - \ell|$ is less than or equal to $\frac{\epsilon}{2}$.

(b) We need to show that there exists an integer $n_2 \in \mathbb{N}^*$ such that for all $n \geq n_2$, the sum $\sum_{k=1}^{n} u_k - n \ell$ divided by $n$ is less than or equal to $\frac{\epsilon}{2}$.

(c) We need to deduce that there exists an integer $n_0 \in \mathbb{N}^*$ such that for all $n \geq n_0$, the absolute difference $|v_n - \ell|$ is less than or equal to $\epsilon$, where $v_n$ is the Cesàro mean.

(d) We will need to conclude the proof using these parts.

If you need me to go through each part in more detail or help with specific steps, please let me know!