Math 220, Midterm 1, February 2002
1 (20 points) Use mathematical induction to prove
that
1
2 + 5 + 8 + · · · + (3n − 1) = n(3n + 1)
2
for n = 1, 2, 3, . . ..
2 (25 points) Use the field axioms and / or any
theorems proved in class to show that 1 > 0. Use
this to show that there is no real number a for which
a2 = −1.
3 (25 points)
√
A. (16 points) Is the set S = {xn : xn = π/n3, n =
1, 2, 3, ...} open, closed, neither, or both as a
subset of the real numbers? What is S 0 ? If
they exist, find sup S, inf S, max S and min S.
Justify your answers.
B. (9 points) Give an example of a closed subset
of the real numbers which is not compact. Be
sure to justify your answer.
4. (30 points) Determine whether the following statements are true or false. Fully justify your
answers.
A. There exists a subset S ⊆ R whose closure is
Q.
B. No subset of the real numbers has exactly 2002
interior points.
1
2
C. If S = {1/m − 1/n : m, n ∈ N} and a, b ∈ R
satisfy −1 < a < b < 1, then there exists
α ∈ S with a < α < b.