Let $X$ be a random variable and $(x_n)_n$ a sequence of $\mathbb{R}^*$ such that $\lim_nx_n=0,\forall k \in \mathbb{N},|\varphi_{X}(x_k)|=1.$
Prove that $X$ is degenerate.
To show that $X$ is degenerate, it's sufficient to prove that $\forall x \in \mathbb{R},|\varphi_X(x)|=1,$ or for two values $p,q$ such that $p/q \in \mathbb{R}-\mathbb{Q},|\varphi_X(p)|=|\varphi_X(q)|=1.$
Do you have any suggestions?
Let $Y$ be independent of $X$ with the same distribution as $X$ and $Z=X-Y$. The characteristic function $g$ of $Z$ is given by $g(t)=|\phi (t)|^{2}$. We have $g(x_n)=1$ for all $n$. This gives $\int [1-\cos (x_nZ)]dP =0$ so $P(x_n Z \in 2\pi \mathbb Z)=1$. This implies that $P((Z=0)\cup |Z| \geq \frac {2 \pi} {|x_n|})=1$. If $N$ is any positive integer then $\frac {2 \pi} {|x_n|} >N$ for some $n$. Hence $P((Z=0) \cup (|Z| >N)=1$. But the intersection of the events $(Z=0) \cup (|Z| >N)$ over all $N$ is $Z=0$ so we get $P(Z=0)=1$. This implies that $X=Y$ a.s. In particular $X$ is independent of itself, so it is almost surely constant.