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.

Let us denote $f(x) = a_0 + a_1 x + a_2 x^2 + \cdot \cdot \cdot + a_n x^n + \cdot \cdot \cdot$.  As $(a_n)$ is a decreasing sequence of positive real numbers, $f$ has a radius of convergence $\geqslant 1$.
We take $z$ such that $\mid z \mid <1$ (hence all the series that will be used are absolutely convergent). We assume $f(z) = 0$.
We have $(1-z)f(z) = 0$, so $a_0 = (a_0 - a_1)z + (a_1 - a_2)z^2 + (a_2 - a_3)z^3 + \cdot \cdot \cdot$
We use triangular inequality : $\mid a_0 \mid = \mid \sum \limits_{n=0}^{+\infty} (a_n - a_{n+1})z^{n+1} \mid \leqslant \sum \limits_{n=0}^{+\infty}\mid a_n - a_{n+1} \mid \cdot \mid z \mid ^{n+1}$.
Moreover, $a_0 > 0$ and $(a_n)$ is decreasing, so :
$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $$a_0 \leqslant \sum \limits_{n=0}^{+\infty} (a_n - a_{n+1})\mid z\mid ^{n+1}$ that is to say $a_0 \leqslant a_0 \mid z \mid - \sum \limits_{n=0}^{+\infty} a_{n+1} (\mid z \mid ^{n+1} - \mid z \mid ^{n+2})$.
$ $
Also $a_n > 0$ and $0 < \mid z \mid$ so this last sum is $>0$. We have finally $a_0 < a_0 \mid z \mid$ but $a_0 > 0$ and $\mid z \mid < 1$. Our assumption was wrong, so $\mid z \mid \geqslant 1$.
