Exploratory Exercises

1. Exploratory Exercises • Suppose that f : AB and g: B C are functions. For each of the following statements decide whether the statement is true (if so, give a proof) or false (if so, give a counterexample). In the cases where the statement is false, decide what additional hypothesis will make the conclusion hold: • If gf is one-to-one, then f is one-to-one. • If gf is one-to-one, then g is one-to-one. • If gf is onto, then f is onto. • If gf is onto, then g is onto.

2. an L means that  > 0  n  d(an, L) < . an L means that  > 0  N  for some n > N, d(an, L) <  . an L means that  N  ,   > 0   n > N, d(an, L) <  . an L means that  N   and  > 0  n > N d(an, L) <  . Making sense of a new definition Students are asked to think of these as “alternatives” to the definition of sequence convergence. Then they are challenged to come up with examples of real number sequences and limits that satisfy the “alternate” definitions but for which an L is false. I usually have the students work on this exercise in class, perhaps with a partner.

3. Intuition and Abstraction • Let X be a metric space, and let A be a subset of X. Suppose that there exist elements a and b of A such that d(a,b) = diam(A). • Prove by giving a counterexample that A need not be closed. • Prove by giving a counterexample that a and be need not lie on the boundary of A. • Prove that if X = n, then a and b must lie on the boundary of A.