Cesàro Theorem

Let $u$ be a sequence of complex numbers that converges toward a complex $l$ . Show that we also have the convergence
$\frac{1}{n}\sum_{k=0}^{n-1}u_{k}\underset{n\to+\infty}{\longrightarrow}l$
Let $f$ be a real continuous function such that $f(x)\underset{x\to+\infty}{\longrightarrow}l$ where $l\in\mathbb{C}$ . Show that:
$\frac{1}{x}\int_{0}^{x}f(t)\,\mathrm{d}t\underset{x\to+\infty}{\longrightarrow}l$
Is there a converse to either of the two previous statements?
In other words, does convergence in the mean always imply the convergence of the sequence or continuous function in question?

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Let $(a_n)_{n \in \mathbb{N}^*}$ be a sequence of complex numbers such that $\lim_{n \to \infty} a_n = \ell \in \mathbb{C}$ . Then:

$\lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^n a_k = \ell$

Setting $\forall n \in \N^{*}, b_n = a_n - \ell$ , we can assume without loss of generality that $\ell = 0$ .

Let $\varepsilon > 0$ . Since $b_n \to 0$ , there exists $N \in \mathbb{N}^*$ such that:

$\forall k > N, \quad |b_k| \le \frac{\varepsilon}{2}$

For any $n > N$ , split the sum into two parts:

$\left| \frac{1}{n} \sum_{k=1}^n b_k \right| \le \frac{1}{n} \sum_{k=1}^N |b_k| + \frac{1}{n} \sum_{k=N+1}^n |b_k| \le \frac{M_N}{n} + \frac{n - N}{n} \frac{\varepsilon}{2} < \frac{M_N}{n} + \frac{\varepsilon}{2}$

where $M_N = \sum_{k=1}^N |b_k|$ is a fixed constant independent of $n$ .

Since $\lim_{n \to \infty} \frac{M_N}{n} = 0$ , there exists $N' \ge N$ such that:

$\forall n > N', \quad \frac{M_N}{n} < \frac{\varepsilon}{2}$

Thus, for all $n > N'$ :

$\left| \frac{1}{n} \sum_{k=1}^n a_k - \ell \right| < \frac{\varepsilon}{2} + \frac{\varepsilon}{2} = \varepsilon$

For the second question, same reasoning.

As counterexamples, you can consider $u_{n}=(-1)^{n}$ and $f=sin$ .

Still, there are some converse results such as when the sequence / function is real and monotonous (or more generally if we already know that it has a limit - finite or not).

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