Let $U_{n}$ and $V_{n}$ be two independent random variables each following the uniform distribution on $\llbracket 1, n \rrbracket$, and $W_{n} = U_{n} \wedge V_{n}$. We admit that, for all $k \in \mathbb{N}^{*}$, the sequence $(\mathbb{P}(W_{n} = k))_{n \in \mathbb{N}^{*}}$ converges to a real number $\ell_{k}$. Show that $$\forall \varepsilon > 0, \quad \exists M \in \mathbb{N}^{*} \text{ such that } \forall m \in \mathbb{N}^{*},\ m \geqslant M \Longrightarrow 1 - \varepsilon \leqslant \sum_{k=1}^{m} \ell_{k} \leqslant 1$$
Let $U_{n}$ and $V_{n}$ be two independent random variables each following the uniform distribution on $\llbracket 1, n \rrbracket$, and $W_{n} = U_{n} \wedge V_{n}$. We admit that, for all $k \in \mathbb{N}^{*}$, the sequence $(\mathbb{P}(W_{n} = k))_{n \in \mathbb{N}^{*}}$ converges to a real number $\ell_{k}$.
Show that
$$\forall \varepsilon > 0, \quad \exists M \in \mathbb{N}^{*} \text{ such that } \forall m \in \mathbb{N}^{*},\ m \geqslant M \Longrightarrow 1 - \varepsilon \leqslant \sum_{k=1}^{m} \ell_{k} \leqslant 1$$