The Cesàro lemma

1. Replace $u_n$ by $u_n-\ell$: we may assume $\ell=0$. Let $\varepsilon>0$ and choose $N$ with $|u_k|<\varepsilon$ for $k>N$. Then
$$\left|\frac{u_1+\dots+u_n}n\right|\le\frac{|u_1|+\dots+|u_N|}n+\frac{n-N}n\varepsilon\le\frac C n+\varepsilon,$$
with $C=\sum_{k\le N}|u_k|$ fixed. For $n$ large, $\frac Cn<\varepsilon$, so the average is below $2\varepsilon$.

2. Write $u_n=u_0+\sum_{k=0}^{n-1}(u_{k+1}-u_k)$. Then
$$\frac{u_n}n=\frac{u_0}n+\frac1n\sum_{k=0}^{n-1}(u_{k+1}-u_k)\longrightarrow0+\ell$$
by part 1 applied to the sequence $u_{k+1}-u_k$.

3. $u_n=(-1)^n$ does not converge, yet its averages are $0$ or $-\frac1n$, which tend to $0$.

Remark. Cesàro averaging "smooths" oscillations: it is the first example of a *summation method* for divergent sequences.