  The Initial Algebra Theorem by Trnkov\'a et al. states, under mild but
inevitable assumptions, that an endofunctor has an initial algebra provided it
has a pre-fixed point. The proof crucially depends on transfinitely iterating
the functor and in fact shows that, equivalently, the (transfinite)
initial-algebra chain stops. We give a constructive proof of the Initial
Algebra Theorem that avoids transfinite iteration of the functor. This is made
possible by using Pataraia's theorem to obtain the east fixed point of a
monotone function on the partial order formed by all subobjects of a given
pre-fixed point. Thanks to properties of recursive coalgebras, this least fixed
point yields an initial algebra. Finally, using transfinite iteration we
equivalently obtain convergence of the initial-algebra chain as an equivalent
condition, overall yielding a streamlined version of the original proof.
