Let's consider sequence $(c_n)$ such that $c_{n + 1} > c_n$ and different numbers $x_0, x_1,..., x_n \in [a, b]$ such that $a + c_1 > 0$. I want to prove that
$$ \det\begin{pmatrix} \frac{1}{x_1 + c_1}, \frac{1}{x_1 + c_2}, \frac{1}{x_1 + c_3},...,\frac{1}{x_1 + c_n} \\ \frac{1}{x_2 + c_1}, \frac{1}{x_2 + c_2}, \frac{1}{x_2 + c_3},...,\frac{1}{x_2 + c_n}, \\ \dots \\ \frac{1}{x_n + c_1}, \frac{1}{x_n + c_2}, \frac{1}{x_n + c_3},...,\frac{1}{x_n + c_n} \end{pmatrix} \neq 0 $$
My work so far
My first idea was to prove it by induction since for $n = 1$ we have determinant equal to $\frac{1}{x_1 + c_1} \neq 0$ and for $n = 2$ we have determinant $\frac{(c_2 - c_1)(x_2 - x_1)}{(x_1 + c_1)(x_2 + c_2)(x_2 + c_1)(x_1 + c_2)} \neq 0 $ since $x_2 \neq x_1$ and $c_2 > c_1$.
However, by assuming our theorem for dimension $n$ I didn't know how to generalize it to dimension $n + 1$. Moreover, I tried to calculate directly this determinant also by induction but I couldn't observe any pattern to assume. Could you please give me a hand in proving this theorem?
