How would you show that this fraction function is convex and decreasing?

Question

Show that

$$ f(\vec{x}) = \frac{1}{x_1 - \frac{1}{x_2 - \frac{1}{x_3 - \frac{1}{x_4}}}} $$

is convex when all denominators are greater than $0$.

Answer 1

$x_4 \mapsto -\frac{1}{x_4}$ is convex for $x_4 < 0$ (for example, use derivative test); $(x_3, x_4) \mapsto x_3 - \frac{1}{x_4}$ is a sum of convex functions, and so is convex for any $x_3$ and $x_4 < 0$; and so on and so forth. Thus your $f$ is convex when all the denominators are less than $0$.