Show that sin((—57Г, 57г)) = [—1,1]. Give examples of bounded open sets A in Ж such that (a) sin A is closed and not open, (b) sin A is open and not closed, (c) sin A is neither open nor closed, (d) sin A is open and closed. (Observe that 0 is automatically bounded.)
The reader may object that we have not yet derived the properties of sin. In my view this does not matter if we are merely commenting on or illustrating our main argument. (I say a little more on this topic in Appendix C.) However, if the reader is interested, she should be able to construct a polynomial P such that (a), (b), (c) and (d) hold for suitable A when sin A is replaced by P(A).
The next exercise gives a simple example of how Lemma 4.31 can be used and asks you to contrast the new 'open set' method with the old 'e, S method
Exercise 4.33. Prove the following result, first directly from Definition 4-28 and then by using Lemma 4-31 instead.
If f : Mm —»• MJ? and g : IRP —»• Ш are continuous, then so is their composition g о f.
(Recall that we write g о f (x) = g(f (x)).)
The reader who has been following carefully may have observed that we have only defined limits of sequences. Here is another notion of limit which is probably familiar to the reader.