Let a random variable $\xi$ take values in $\mathbb Z_+$ and $\mathbb E \xi \lt \infty $. How do I prove that the characteristic function $\psi _\xi$ is bounded and continuous on $[0,1].$
My Attempt: The discrete RV characteristic function looks like $$\psi_\xi (t) = \sum_{k=1}^{\infty} e^{itk} P(\xi= k).$$ So taking values from $\mathbb Z_+$ where $x_1 \leq x_2$ then $x_{1}^k \leq x_{2}^k$ for all $k\in \mathbb Z_+$. The function $\psi_\xi (x)$ can be written as: $$\psi_\xi (x_1) = \sum_{k=0}^{\infty} x_{1}^k p(\xi=k) \leq \sum_{k=0}^{\infty} x_{2}^k p(\xi=k) = \psi_\xi (x_2).$$ Therefore the function $\psi_\xi (x)$ cis an increasing function. But I am stuck here, how do I show that it is bounded and continuous on $[0,1]$. Any help or hint is appreciated. Thanks in Advance!