  Let $H^\infty(\Delta)$ be the uniform algebra of bounded analytic functions
on the open unit disc $\Delta$, and let $\mathfrak{M}(H^\infty)$ be the maximal
ideal space of $H^\infty(\Delta)$. By regarding $\Delta$ as an open subset of
$\mathfrak{M}(H^\infty)$, the corona problem asks whether $\Delta$ is dense in
$\mathfrak{M}(H^\infty)$, which was solved affirmatively by L. Carleson.
Extending the cluster value theorem to the case of finitely many functions, we
provide a direct proof of the corona theorem: Let $\phi$ be a homomorphism in
$\mathfrak{M}(H^\infty)$, and let $f_1, f_2, \dots, f_N$ be functions in
$H^\infty(\Delta)$. Then there is a sequence $\{\zeta_j\}$ in $\Delta$
satisfying$f_k(\zeta_j) \rightarrow \phi(f_k)$ for $k=1, 2, \dots, N$. On the
other hand, the corona problem remains unsolved in many general settings, for
instance, certain plane domains, polydiscs and balls, our approach is so
natural that it may be possible to deal with such cases from another point of
view.
