  We construct a nontrivial inverse system of Cuntz algebras $\{{\cal
O}_{n}:2\leq n<\infty\}$, whose inverse limit is *-isomorphic onto ${\cal
O}_{\infty}$. By using this result, it is shown that the $K_{0}$-functor is
discontinuous with respect to the inverse limit even if the limit is a
C$^{*}$-algebra.
