One usual "proof" of <a href="http://en.wikipedia.org/wiki/Leopoldt%27s_conjecture"> Leopoldt Conjecture </a> is that $\mathbb{Z}_p$ is $\mathbb{Z}$-flat, hence the rank of the $p$-adic completion of the units of a number field has the same rank of the units themselves (which is Leopoldt Conjecture) because you can obtain the completion simply as $\mathcal{O}^\times\otimes\mathbb{Z}_p$.