1. Let a,b âˆˆ â„ with a â‰ 0, b â‰ 0. Prove that (ab)â»Â¹ = aâ»Â¹bâ»Â¹.
2. Let q âˆˆ â„š and x âˆˆ â„ - â„š. Prove that q + x âˆˆ â„ - â„š.
3. Using basic set notation, do the following:
a. Write â„¤ in terms of â„• and 0
b. Write â„š in terms of â„¤
c. Write the irrationals in terms of â„ and â„š
4. There exists real numbers a, b, c such that a â‰¤ b, 0 < c and ac â‰¤ bc.
True or False
5. Indicate the least upper bound of the following sets. If a set does not have a least upper bound, just write â€œnone.â€
a. (2,3)
b. (2,3]
c. â„š
d. â„š âˆ© (1,Ï€]
6. Let a âˆˆ â„ and let A âˆˆ (0,âˆž) with x âˆˆ (a - A, a + A). Then there exists some P âˆˆ (0,A) such that x âˆˆ (a - P, a + P).
True or False
7. Prove each of the following using induction:
a. Let a âˆˆ â„ and a â‰¥ 1. Prove an â‰¥ a for all n âˆˆ â„•.
b. Prove that 1/2 + 1/4 + 1/8 + â‹¯ + 1/2â¿ = 1 - 1/2â¿ for all n âˆˆ â„•.

1. Let a,b âˆˆ â„ with a â‰ 0, b â‰ 0. Prove that (ab)â»Â¹ = aâ...

1. Let a,b âˆˆ â„ with a â‰ 0, b â‰ 0. Prove that (ab)â»Â¹ = aâ...