We assume that there exists $t _ { 0 } \in \mathbb { R } ^ { * }$ such that $\left| \phi _ { X } \left( t _ { 0 } \right) \right| = 1$. We assume further that $X ( \Omega )$ is countable and we use the notations of question 2 (i.e. $X ( \Omega ) = \left\{ x _ { n } , n \in \mathbb { N } \right\}$ with $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$). Using the result of Q11, deduce that $\sum _ { n = 0 } ^ { + \infty } a _ { n } \left( 1 - \cos \left( t _ { 0 } x _ { n } - t _ { 0 } a \right) \right) = 0$.
We assume that there exists $t _ { 0 } \in \mathbb { R } ^ { * }$ such that $\left| \phi _ { X } \left( t _ { 0 } \right) \right| = 1$. We assume further that $X ( \Omega )$ is countable and we use the notations of question 2 (i.e. $X ( \Omega ) = \left\{ x _ { n } , n \in \mathbb { N } \right\}$ with $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$).\\
Using the result of Q11, deduce that $\sum _ { n = 0 } ^ { + \infty } a _ { n } \left( 1 - \cos \left( t _ { 0 } x _ { n } - t _ { 0 } a \right) \right) = 0$.