Let $\Omega$ be a bounded open subset of $\mathbb{R}^n$. Let $f,g\in C(\overline{\Omega},\mathbb{R}^n)$ and $b\in \mathbb{R}^n$. Then why we have $$ dist(b,g(\partial\Omega))\geq dist(b,f(\partial\Omega))-\Vert g-f\Vert $$ where $dist$ is the distance between a point and a set in $\mathbb{R}^n$ and $$ \Vert f\Vert=\sup_{x\in\overline{\Omega}}|f(x)| $$ with $\vert\cdot\vert$ the usual Euclidean norm in $\mathbb{R}^n$.
Thank you very much.
$$d(b,g(\partial \Omega))=\inf_{x\in \partial \Omega}|b-g(x)|;\quad d(b,f(\partial \Omega))=\inf_{x\in \partial \Omega}|b-f(x)| $$ Triangle inequality: $$|b-g(x)|\geq |b-f(x)|-|g(x)-f(x)|\qquad \forall x\in\partial \Omega $$ Take the $\inf$ over $x\in \partial \Omega$ on both terms: \begin{align*}d(b,g(\partial \Omega))&\geq d(b,f(\partial \Omega))-\inf_{x\in \partial \Omega}|g(x)-f(x)|\geq d(b,f(\partial \Omega))-\sup_{x\in \partial{\Omega}}|g(x)-f(x)|\geq \\ &\geq d(b,f(\partial \Omega))-\sup_{x\in \overline{\Omega}}|g(x)-f(x)|=d(b,f(\partial \Omega))-\|g-f\| \end{align*}