In their classic paper "Class fields of abelian extensions of $\mathbf{Q}$", Mazur and Wiles assert that

"in a cyclotomic $\mathbf{Z}_p$-extension only finitely many primes lie above any prime of $\mathbf{Q}$."

My only other source in learning this material so far has been Washington's "Introduction to Cyclotomic Fields", and the only result along these lines is that such extensions are unramified outside of $p$.

So apparently, all primes lying above $l \neq p$ stop splitting at some finite level $K_n$, after which they remain inert. I've been unable to make much progress is proving this.

How can we see that this statement is true, and what other, more general results do we have about prime decomposition in cyclotomic extensions?