1. Indicate the least upper bound of the following sets. If a set does not have a least upper bound, just write "none."

a. (2,3)

b. (2,3]

c. Q

d. Q∩ (1, π]

2. Prove each of the following using induction:

1. Let a∈R and a ≥ 1. Prove an ≥ a for all n∈N.

2. Prove that 1/2 + 1/4 + 1/8 + ? + 1/2^{n} = 1 - 1/2^{n} for all n∈N.

3. Using basic set notation, do the following:

a. Write Z in terms of N and 0

b. Write Q in terms of Z

c. Write the irrationals in terms of R and Q

4. There exists real numbers a, b, c such that a ≤ b, 0 < c and ac ≤ bc. (T/F)

5. Let a, b∈R with a ≠ 0, b ≠ 0. Prove that (ab)-1 = a^{-1}b^{-1}

6. Let q∈Q and x∈R - Q. Prove that q + x∈R - Q.

7. Let a∈R and let A∈ (0, ∞) with x ∈ (a - A, a + A). Then there exists some P∈ (0, A) such that x ∈ (a - P, a + P). (T/F)

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