Does $g$ have a minimum on $X$?

Question

Let $f$ and $g$ be two functions from $X$ to $\mathbb{R}$ such that $f=\frac{1}{g}$

Suppose $f$ is continuous and not bounded and $g>0$ and that there exists a sequence $u_{n}$ of $X$ such that $g(u_{n})$ converges to zero

Does $g$ have a minimum ?

Answer 1

If $g>0$ and $g(u_n) \to 0$ then the infimum of $g$ is $0$. Obviously the infimum is not attained.

Answer 2

The answer is not necessarily. For instance, consider the function $f:(0,1)\to \Bbb R$ given by $$ f(x) = \frac{x}{1-x}. $$