prove the following compact result about $GL(n,mathbb{R})$

Question

Let $K$ be a compact subset of $mathbb{R}^n$:
prove for any $Ain GL(n,mathbb{R})$ (endowed with Euclidean topology),there exist some neighborhood $U(A)subset GL(n,mathbb{R})$ near $A$ and some compact set $Fsubset mathbb{R}^n$(which only depends on choice of $A$) . such that $U(A)K$ always contains in $F$.
My attemp since ${A}$  is compact in $GL(n,mathbb{R})$,there always exist some $U(A)$ as neiborhood of $A$ that is relative compact in $GL(n,mathbb{R})$.We only need to show for set of relative compact matrix if acts on compact set $K$ it is still contains in compact set.

stlinex · Accepted Answer

Let $U(A)={Bin GL(n,mathbb R)mid frac{||A||}{2}<||B||<2||A||}$, where $||cdot||$ is the operator norm.
($U(A)$ is open follows from the fact that all norm on $mathbb R^{ntimes n}$ are equivalent.)
Let $R=max{||x||mid xin K}<infty$.
Then $||Bx||leq ||B||cdot ||x||leq 2||A||cdot  R$ for all $Bin U(A)$ and $xin K$.
Take $F={xin mathbb R^nmid ||x||leq 2||A||cdot R}.$

prove the following compact result about $GL(n,mathbb{R})$

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