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