TITLE: Inverse limits of groups of units of a ring.
QUESTION [7 upvotes]: Can someone enlighten me on the following isomorphism described in "Exercises in Modern Algebra" by Professor A. Hattori (in Japanese)?

Let $\lbrace R^{\mu} \rbrace$ be an inverse system of rings (with units.)
  Then $(\varprojlim (R^{\mu}))^{\times} \cong \varprojlim (R^{\mu})^{\times}$ holds where $^{\times}$ denotes the group of units.
  The verification of this fact is easy so the proof is omitted.

I was not able to come up with a proof, so I googled and found one by Professor Igusa (Corollary 2.7. of http://people.brandeis.edu/~igusa/Math101aF07/Math101a_notesBall.pdf).
I know I have to be satisfied with this, but I don't think the proof is "easy": the techniques used hardly fit into first-year algebra courses.
Is there an easy trick that Professor Hattori had in mind which I am overlooking?

REPLY [5 votes]: Whether this counts as an "easy trick" is up to you, but the group of units is a functor $\text{Ring} \to \text{Grp}$ which has a left adjoint, namely the group ring construction. Any functor with a left adjoint preserves limits.