Let $(a_n)_{n \geq 0}$ be a strictly decreasing sequence of positive real numbers , and let $z \in \mathbb C$ , $|z| < 1$. Let $(a_n)_{n \geq 0}$ be a strictly decreasing sequence of positive real numbers , and let $z \in \mathbb C$ , $|z| < 1$. Prove that the sum $a_0 + a_1z + a_2z^2 + \cdots + a_nz^n +\cdots $ is never equal to zero.

Simply note that $(1-z)\sum_{k=0}^\infty a_kz^k=a_0+\sum_{k=1}^\infty (a_k-a_{k-1})z^k$.
If $\sum_{k=0}^\infty a_kz^k=0$ for some $|z|<1$, $$a_0=\sum_{k=1}^\infty (a_{k-1}-a_k)z^k\implies$$ $$ \begin{align} a_0\leq \left| \sum_{k=1}^\infty (a_{k-1}-a_k)z^k\right| &\leq \sum_{k=1}^\infty (a_{k-1}-a_k)|z|^k \\ &< \sum_{k=1}^\infty (a_{k-1}-a_k)\\ &= a_0 - \lim_n a_n \end{align}$$
Hence $\lim a_n<0$ which is a contradiction.
This has some interesting consequences, like $\frac{1}{1-z}+e^z$ has no zeroes if $|z|<1$...
