  We reprove the $\lambda$-Lemma for finite dimensional gradient flows by
generalizing the well-known contraction method proof of the local (un)stable
manifold theorem. This only relies on the forward Cauchy problem. We obtain a
rather quantitative description of (un)stable foliations which allows to equip
each leaf with a copy of the flow on the central leaf -- the local (un)stable
manifold. These dynamical thickenings are key tools in our recent work [Web].
The present paper provides their construction.
