1. Suppose a set
is a dense subset of another set
and that both are subsets of a metric space
. Which of the following is true?

2. Consider a sequence of functions
, which are contraction mappings, i.e. for all
and for all
,
. Let
be an open, connected subset of
. Suppose that
uniformly on S and
pointwise on
. Which of the following is true?

3. Suppose
is a compact subset of
. Which of the following is false?

4. Consider the following (false) claim and 'proof;' find the incorrect step. Claim: All positive integers are equal. Proof:
1. It suffices to show that for any two positive integers, A and B, A = B.
2. Further, it suffices to show that for all N > 0, if A and B are positive integers which satisfy MAX(A, B) = N, then A = B.
3. (Base Case) If N = 1, then A and B, being positive integers, must both be 1. So A = B.
4. (Induction Step) Assume that the theorem is true for some value k. Take positive integers A and B with MAX(A, B) = k+1. Then, MAX((A-1), (B-1)) = k.
5. Therefore, by induction, (A-1) = (B-1). Consequently, A = B. Hence, A = B for all positive integers A and B by induction. Q.E.D.

5. Let
be differentiable and let
and
(in particular these limits exist). Which of the following implies the existence of
for which
?

6. The Rational Numbers have the same cardinality as which of the following?

7. Which of the following is NOT a norm on the set of continuous functions on
?

8. Let
be a sequence in
. Which of the following is NOT sufficient to guarantee that
is convergent?

9. Consider a sequence
. What does it mean to say that
?

10. Let
be continuously differentiable and satisfying, for all
, the inequality
. Which of the following is true?

11. The well-ordering principle of the natural numbers asserts which of the following?

12. Which of the following is sufficient to imply that a series
converges?

13. Consider two subsets
. Recall that the symmetric difference of two sets
and
is
. What is the complement of this set in
?

14. A metric space
is complete under which condition?

15. Suppose that
is monotone increasing. Which of the following is false?

16. Let
be differentiable and define
by the formula
. Which of the following is the value of
?

17. What does it mean to say that a function
is differentiable on its domain with derivative
?

18. Suppose that
is a complete metric space, and let
. Which of the following is true?

19. A set
is a perfect set if it is closed and if every element of
is an accumulation point. Which of the following is true?

20. Suppose that
and
are elements of a real inner product space
such that
and
. What can we say about
?

21. Suppose
.If
, which of the following statements is true?

22. Suppose a function
is uniformly continuous. Then, which of the following is true?

23. If
is a continuous function, and
is compact, which of the following is true?

24. Given a function
, and two subsets
, what does one mean by
?

25. Suppose that a function
is monotone. Recall that
has the Darboux property on a set
, if and only if given any
, for any
,
. Let
be an interval. Which of the following is true?