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
Making sense of a new definition

• an ? L means that ? ? gt 0 ? n ?? ? d(an , L)
  lt ?.
• an ? L means that ? ? gt 0 ? N ?? ? for some n
  gt N, d(an , L) lt ? .
• an ? L means that ? N ? ?, ? ? gt 0 ? ? n gt N,
  d(an , L) lt ? .
• an ? L means that ? N ? ? and ? ? gt 0 ? n gt N ?
  d(an , L) lt ? .

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.