Let $M$ and $N$ be smooth manifolds. There are different topologies we can equip the space $C^\infty (M, N)$ of smooth mappings between them with. Two of them are the compact-open topology and the Whitney $C^k$-topologies for $k \in \mathbb{N} \cup \{\infty\}$ (I'm using this definition based on jet bundles).
I have read claims that when $M$ is compact these agree. Is this true? Could someone provide a proof or a reference?
EDIT: I'm starting to seriously doubt that this is true for $k \geq 1$. I can show that the Whitney topology is finer than the compact-open, but for say $f: \mathbb{S}^1 \to \mathbb{R}$ allowing to change the value by any $\varepsilon$ on a compact set with nonempty interior is sufficient to make the derivative arbitrarily large.