For a compact set $K\subset \mathbb M_n(\mathbb R)$, the eigenvalues of matrices in $K$ form a bounded set Let $K\subset \mathbb M_n(\mathbb R)$ be a compact subset. Then I have to show that :


All the eigen values of the elements of $K$ form a bounded set.


My work: consider the map $K \to det K$ which is continuous. Image set is compact in $\mathbb C$, hence closed bounded. If $\lambda_i (i=1\ldots,n)$ are the eigenvalues, then $\det K=\prod \lambda_i$ is bounded which in turn  gives $\lambda_i$ bounded.
Is my approach correct? Is there any better way to do it?

Let $||*||$ be any norm on $ \mathbb R^n$ and let $||*||_O$ the matrix norm induced by $||*||$
Since $K$ is compact, $K$ is bounded. Thus, there is $c>0$ such that
$||A||_O \le c$ for all $A \in K$.
Now let  $A \in K$ and let $ \lambda$ be an eigenvalue of $A$. Then there is $x \in \mathbb R^n$ with $Ax= \lambda x$ and $||x||=1$.
We get
$$| \lambda|=|| \lambda x||= ||Ax|| \le ||A||_O*||x||=||A||_O \le c.$$
