Reference no: EM131034752
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 + ? + 1/2n = 1 - 1/2n 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-1b-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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