Question:
For the space $C^{k}(U)$ of $k$ times differentiable functions on $U$, where $U \subset \mathbb{R}^n$, and with topology induced by the family $||f||_{\alpha,K}=\sup\limits_{x \in K}\big|D^{\alpha}f(x)|$ of seminorms, $K \subset U$ compact and $|\alpha|\leq k$, how can I show I show that there is a countable family of such seminorms that gives the same topology on $C^{k}$?
Background:
Im studying a document by Stephen Semmes, and having introduced the sup-seminorm topology on $C^{k}$, he'd like to show that there is a countable family of such seminorms that gives the same topology as that induced by the family $||f||_{\alpha,K}=\sup\limits_{x \in K}\big|D^{\alpha}f(x)|$ of seminorms.
He starts (page 4, "countably many seminorms") with $\mathbb{R}^n$, equip it with the euclidian norm and reminds the reader that any compact set can be contained in the closed ball $\overline{B}(0,r)$ for some $r > 0$. In particular, he stresses the fact that the closed balls $\overline{B}(x,r)$, $x \in \mathbb{R}^n$, are contained in some closed ball centred at $0$. He then says the following:
"This implies that one can get the same topology on $C(\mathbb{R}^n)$ using the sequence of supremum seminorms associated to the closed balls $\overline{B}(0,r)$ with $r \in \mathbb{Z}_+$"
He then continues the argument for $C^{k}$ which is what I'm after.
I cannot see the implication, and I don't know where I should to turn to fill my gap. I'd be very thankful for an explanation and how the latter is implied by the former.