From chapter 1 we have the following definition of space $\mathcal{D}_K$.
If $K$ is a compact set in $\mathbb{R}^n$ then $\mathcal{D}_K$ denotes the space of all $f \in C^{\infty}(\mathbb{R}^n)$ whose support lies in $K$.
The topology on such space is introduced by the family of seminorms
$$ p_N(f) = \max \left\{ \left| D^{\alpha} f(x) \right| : x \in K_N, \left| \alpha\right| \leq N\right\} $$
I was wondering if $n = 1$ it is possible to prove that $\mathcal{D}_K$ is bounded w.r.t. to the topology induced by the family $\left\{ p_N \right\}$.
The family of seminorms above defines a local base
$$ V_N = \left\{f \in C^{\infty}(\mathbb{R}) : N p_N(f) < 1 \right\} $$ for a locally convex space (the closure of $V_N$ is a convex set). Therefore if for each $N$ there's an $s_N > 0$ such that
$$ \mathcal{D}_K \subset t V_N \;\;, \;\; t > s_N $$
Then the set is bounded. Defining q_N = N p_N, my attempt to prove this (assuming this is true) was to use the Minkowsky functional on $V_N$ and define $s_N$ as
$$ s_N = \sup \left\{ \mu_{V_N}(\mathcal{D}_K) \right\}, $$
where
$$ \mu_{V_N}(f) = \inf \left\{ t > 0 : t^{-1}f \in V_N \right\} = \inf \left\{ t > 0 : f \in tV_N \right\} $$
If I can prove such $\sup$ exists I think this is all I need. I do struggle however to prove this, can anyone help me or give me a clue on how to do this?
Thank you