So, let me get as far as I can and see if either A) it works or B) I run into problems and give up.
Let and let .
Since, is continuous we may say that and so . But, it's pretty easy to show that and so and so and so and also since is continuous we see that , but it is fairly easy to see that and so . Thus, .
That said, I don't think (I haven't tried) it would be very hard to show that it converges.
EDIT: Embarrassing as it is to say, it isn't even obvious to me what "converge" would mean here. What metric are we talking about with or I guess more properly ? The only thing that immediately comes to mind is the one induced by
For example, the Lie algebra of the unit circle is the imaginary line (it's the tangent space to the identity), and the exponential takes the imaginary line to the unit circle.