Let $X$ be a Bernoulli random variable with parameter $\lambda \in ]0,1[$. Show that $$d_{VT}\left(p_X, \pi_\lambda\right) = \lambda\left(1 - e^{-\lambda}\right).$$ Deduce that $$d_{VT}\left(p_X, \pi_\lambda\right) \leq \lambda^2.$$
Let $X$ be a Bernoulli random variable with parameter $\lambda \in ]0,1[$. Show that
$$d_{VT}\left(p_X, \pi_\lambda\right) = \lambda\left(1 - e^{-\lambda}\right).$$
Deduce that
$$d_{VT}\left(p_X, \pi_\lambda\right) \leq \lambda^2.$$