embeddings of TVS are continuous

Question

How to show that the two embeddings of TVS are continuous:

$C^{\infty}_c\subset S\subset C^{\infty}$?

According to Wikipedia, the definition of "Continuously embedded" is that " one normed vector space is said to be continuously embedded in another normed vector space if the inclusion function between them is continuous". Is this the correct idea to follow?

EDIT: $S$ is the Schwartz space, $C^{\infty}$ is the space of smooth functions, and $C^{\infty}_c$ is the space of smooth functions whose support are compact.

The topology on $S$ is the normed topology:

And I really do not know what topology we should impose on the other 2 spaces...

Thanks!