Using the previous notation we have
hence
is a decreasing function, hence
is a increasing. Now suppose that
then since f(f(a)) is increasing we have
and hence
. Similairly we get that if
then
. So we have shown that the sequences a_1,a_3... and a_2,a_4... are monotonic (don't go up and down) hence they both have a limit (they are bounded). Now it only suffices to find both of these limits by solving the equation f(f(a))=a (you get this equation by "taking the limit of both sides" in the reccurence and of course using the fact that f(a) is continous) and showing that the two limits are indeed the same, hence the whole sequence converges.
too bad no latex, may be an annoying read