By Daniel A. Murray

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6. (i) The set Q is countable. (ii) The set R is uncountable. Proof. (i) Set A1 = {0} and for each n ≥ 2, let An be the set given by An = ± p : where p, q ∈ N are in lowest terms with p + q = n . q The first few of these sets look like A1 = {0} , A4 = 1 −1 3 −3 , , , 3 3 1 1 A2 = , 1 −1 , 1 1 and , A5 = A3 = 1 −1 2 −2 , , , 2 2 1 1 , 1 −1 2 −2 3 −3 4 −4 , , , , , , , 4 4 3 3 2 2 1 1 . The crucial observation is that each An is finite and every rational number appears in exactly one of these sets.

8. Let B be a set of positive real numbers with the property that adding together any finite subset of elements from B always gives a sum of 2 or less. Show B must be finite or countable. 9. A real number x ∈ R is called algebraic if there exist integers a0 , a1 , a2 , . . , an ∈ Z, not all zero, such that an xn + an−1 xn−1 + · · · + a1 x + a0 = 0. Said another way, a real number is algebraic if it is the root of a polynomial with integer coefficients. Real numbers that are not algebraic are called transcendental numbers.

In the next section, we will prove that α2 = 2. 1, this implies α is not a rational number. If we are restricting our attention to only rational numbers, then α is not an allowable option for sup S, and the search for a least upper bound goes on indefinitely. Whatever rational upper bound is discovered, it is always possible to find one smaller. 6 depend on results about how Q and N fit inside of R. These are discussed in the next section. In the meantime, it is possible to prove some intuitive algebraic properties of least upper bounds just using the definition.

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