Suppose we have to non-negative functions $f$ and $g$ and we want to bound the difference of their infimums \begin{align} \left|\inf_{x\in \mathbb{R}}f(x)-\inf_{x\in \mathbb{R}}g(x)\right| \end{align} How can that be done?
Of course one can use triangle inequality to show that \begin{align} \left|\inf_{x\in \mathbb{R}}f(x)-\inf_{x\in \mathbb{R}}g(x)\right| \le \left|\inf_{x\in \mathbb{R}}f(x) \right|+\left|\inf_{x\in \mathbb{R}}g(x)\right|=\inf_{x\in \mathbb{R}}f(x) +\inf_{x\in \mathbb{R}}g(x) \end{align}
However, this bound is to loose.
I was wondering whether we can get a bound of the form
\begin{align} \left|\inf_{x\in \mathbb{R}}f(x)-\inf_{x\in \mathbb{R}}g(x)\right| \le \inf_{x\in \mathbb{R}}\left|f(x)-g(x)\right| \end{align}
If not, perhaps we can get a bound of the form
\begin{align} \left|\inf_{x\in \mathbb{R}}f(x)-\inf_{x\in \mathbb{R}}g(x)\right| \le K\inf_{x\in \mathbb{R}}\left|f(x)-g(x)\right| \end{align} where $K$ is some constant.