If $X$ is a metric space and $f$ and $g$ are functions from $X$ to $\mathbb{R}$, $\displaystyle \sup_{x \in X} |f(x)|$ and $\displaystyle \inf_{x \in X} |g(x)|$ exists, and we have that $\displaystyle \sup_{x \in X} \frac{|f(x)|}{|g(x)|} \leq 1$ where $g(x) \neq 0$, can we conclude that $\displaystyle \sup_{x \in X} \frac{|f(x)|}{|g(x)|} = \frac{\displaystyle \sup_{x \in X} |f(x)|}{\displaystyle \inf_{x \in X} |g(x)|}$? Can we therefore conclude that $\displaystyle \sup_{x \in X} |f(x)| \leq \inf_{x \in X} |g(x)|$?
Thank you.