  We carefully present an elementary proof of the well known theorem that each
homotopy group (or, in degree zero, pointed set) of the inverse limit of a
tower of fibrations maps naturally onto the inverse limit of the homotopy
groups (or, in degree zero, pointed sets) of the spaces in the tower, with
kernel naturally isomorphic to $\lim^{1}$ of the tower of homotopy groups of
one dimension higher.
