  The Initial Algebra Theorem by Trnkov\'a et al.~states, under mild
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. For a given pre-fixed point $A$ of
the functor, it uses Pataraia's theorem to obtain the least fixed point of a
monotone function on the partial order formed by all subobjects of $A$. Thanks
to properties of recursive coalgebras, this least fixed point yields an initial
algebra. We obtain new results on fixed points and initial algebras in
categories enriched over directed-complete partial orders, again without
iteration. 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.
