Let $U$ be a binomial random variable with parameters $n \in \mathbf{N}^*$ and $\lambda \in ]0,1[$. Prove the inequality $$d_{VT}\left(p_U, \pi_{n\lambda}\right) \leq n\lambda^2.$$

Let $\alpha$ be a strictly positive real number. For all natural number $n$ such that $n > \lfloor \alpha \rfloor$, we denote by $B_n$ a binomial random variable with parameters $n$ and $\frac{\alpha}{n}$. For all natural number $k$, determine $$\lim_{n \rightarrow +\infty} P\left(B_n = k\right)$$ One may use the previous question.

Let $\alpha$ and $\beta$ be two strictly positive real numbers. Using the results and methods above, show that $$d_{VT}\left(\pi_\alpha, \pi_\beta\right) \leq |\beta - \alpha|.$$