I am trying to prove the following result:
"Let $f:[a,b] \to [c,d]$ be a homeomorphism on the indicated intervals. Prove that $f$ maps endpoints to endpoints."
This problem follows the section in which it is proved that the connected subsets of $\mathbb{R}$ are exactly the intervals. I have really not made much progress. Can someone give me an idea of how to prove this?
Additionally, can you give me a sense of what your initial thoughts were while starting to think about the problem? I'm very interested not only in how to prove the statement, but in the thought process involved. And an attempt at generalization to any problem is most welcome (I may try to make this sort of idea into its own question.)