  Sz.-Nagy and Foias proved that each $C_{\cdot0}$-contraction has a dilation
to a Hardy shift and thus established an elegant analytic functional model for
contractions of class $C_{\cdot0}$. This has motivated lots of further works on
model theory and generalizations to commuting tuples of
$C_{\cdot0}$-contractions. In this paper, we focus on doubly commuting
sequences of $C_{\cdot0}$-contractions, and establish the dilation theory and
the analytic model theory for these sequences of operators. These results are
applied to generalize the Beurling-Lax theorem and Jordan blocks in the
multivariable operator theory to the operator theory in countably infinitely
many variables.
