Recall that a random variable $X$, taking values in $\mathbb{N}$, follows the Poisson distribution $\mathcal{P}(\lambda)$ with parameter $\lambda$ if, for all $n \in \mathbb{N}$: $$\mathrm{P}(X = n) = \frac{\lambda^{n}}{n!} \mathrm{e}^{-\lambda}$$ Let $\left(X_{n}\right)_{n \geqslant 1}$ be a sequence of mutually independent random variables, with distribution $\mathcal{P}(\lambda)$, $S_{n} = X_{1} + X_{2} + \cdots + X_{n}$, and $T_{n} = \frac{S_{n} - n\lambda}{\sqrt{n\lambda}}$. Show that, for all $\varepsilon > 0$, there exists a real number $c(\varepsilon)$ such that, if $c \geqslant c(\varepsilon)$ and $n \in \mathbb{N}^{*}$, we have $\mathrm{P}\left(\left|T_{n}\right| \geqslant c\right) \leqslant \varepsilon$.
Recall that a random variable $X$, taking values in $\mathbb{N}$, follows the Poisson distribution $\mathcal{P}(\lambda)$ with parameter $\lambda$ if, for all $n \in \mathbb{N}$:
$$\mathrm{P}(X = n) = \frac{\lambda^{n}}{n!} \mathrm{e}^{-\lambda}$$
Let $\left(X_{n}\right)_{n \geqslant 1}$ be a sequence of mutually independent random variables, with distribution $\mathcal{P}(\lambda)$, $S_{n} = X_{1} + X_{2} + \cdots + X_{n}$, and $T_{n} = \frac{S_{n} - n\lambda}{\sqrt{n\lambda}}$. Show that, for all $\varepsilon > 0$, there exists a real number $c(\varepsilon)$ such that, if $c \geqslant c(\varepsilon)$ and $n \in \mathbb{N}^{*}$, we have $\mathrm{P}\left(\left|T_{n}\right| \geqslant c\right) \leqslant \varepsilon$.