  We show that in the case of positively-curved Friedmann-Lema\^{\i}tre
universes $(k=+1)$, an inflationary period in the early universe will for most
initial conditions not solve the horizon problem, no matter how long inflation
lasts. It will only do so for cases where inflation starts in an almost static
state, corresponding to an extremely high value of $\Omega_{\Lambda}$,
$\Omega_{\Lambda} \gg 1$, at the beginning of inflation. For smaller values, it
is not possible to solve the horizon problem because the relevant integral
asymptotes to a finite value (as happens also in the de Sitter universe in a
$k=+1$ frame). Thus, for these cases, the causal problems associated with the
near-isotropy of the Cosmic Background Radiation have to be solved already in
the Planck era. Furthermore both compact space sections and event horizons will
exist in these universes even if the present cosmological constant dies away in
the far future, raising potential problems for M-theory as a theory of gravity.
