Recently, a friend of mine, Chris, posed the following question to me:
Consider the sequence of functions For what values does the limit of this sequence exist, and what is that limit?
After a few moments, it is relatively easy to convince oneself that for all , this sequence converges to , but a complete proof seemed tedious. Chris then told me to consider the concept of fixed points and a simple solution would arise.
If such a sequence were to converge to a limit, then it could only do so at a fixed point of that sequence, i.e. a point such that , and in that case, the limit would be . What are the fixed points of the composition? Only ! Then it takes only the simple exercise to see that the sequence does in fact have a limit for every x (one might split the cases for positive and negative angles, in which case one has a decreasing/increasing sequence that is bounded below/above for example).
A cute little exercise, I think.