Let be a complete metric space. A contraction is a function for which there exists some constant such that for all .
Answer the following yes-no questions:
If is a contraction, does have a fixed point? I.e., is there some such that
If has a fixed point, is a contraction?
Hint: The first question is much harder than the second. In fact, the answer is yes, and this extremely important result is known as the Banach fixed point theorem.
To prove it, choose an arbitrary and set for . Then, show that converges to some and that is the desired fixed point.