  Invoking the density argument of Dundas-Goodwillie-McCarthy, we extend the
Fundamental Theorem of $K$-theory from the category of simplicial rings to the
category of $\mathbb{S}$-algebras. As an intermediate step, we prove the
Fundamental Theorem for simplicial rings appealing to recent results from the
first author's thesis. This recovers as a special case the Fundamental Theorem
for the $K$-theory of spaces appearing in
H\"uttemann-Klein-Vogell-Waldhausen-Williams.
