KMB, 10/11/99

M1F Foundations of Analysis—Problem Sheet 6

1) Write down (no explanation necessary) the least upper bounds and greatest lower bounds of the sets below, or say “none” if they don’t exist.

a) {−1, −2, −3} b) {−1, −2, −3, …}

c) {x ∈ ℚ : x^{2} < 2 } d) {x ∈ ℚ : x^{2} > 2 }

e) {x ∈ ℝ : x is irrational, and x^{2} < 2 }

2) Give examples of:

a) A set of rationals with a least upper bound that is irrational.

b) A set of irrationals with a least upper bound that is rational.

c) A set of rationals with a rational upper bound but an irrational lower bound.

3) Write down proofs of the following. Both proofs should be only a few lines long.

a) If A ⊆ B are sets of reals, then a lower bound for B is also a lower bound for A.

b) If A ⊆ B are sets of reals, and the greatest lower bound of A is x, and the greatest lower bound of B is y, then y ≤ x.

4) Give proofs of the following two steps that I skipped during the proof of the least upper bound theorem:

a) Let c be a real number. Let S be a set of reals, with greatest lower bound x. Let T be the set that you get by “shifting everything in S up by c”. That is, T = {x ∈ ℝ : x − c ∈ S }. Prove that x + c is a greatest lower bound for T. [Don’t just say “it’s obvious”—but actually check that x + c satisfies the definition of a greatest lower bound for T.]

b) Let S be a set of reals, and let T denote the set “−S”. More formally, set T = {x ∈ ℝ : −x ∈ S }. Prove that if S has a greatest lower bound, x, then −x is a least upper bound for T.

5) Let x_{1}, x_{2}, x_{3}, … be a sequence of real numbers. For any integer n ≥ 1, define the set T_{n} to be the set {x_{n}, x_{n+1}, x_{n+2}, … }. So for example,

T_{1} = {x_{1}, x_{2}, x_{3}, x_{4}, … }

and

T_{2} = {x_{2}, x_{3}, x_{4}, … },

and so on.

Assume that T_{1} has a lower bound. Deduce that T_{n} has a greatest lower bound for all n ≥ 1. Call this lower bound b_{n}. Show that b_{1} ≤ b_{2} ≤ b_{3} . . ..

Sometimes the set {b_{1}, b_{2}, b_{3} … } has an upper bound. For the sequences (x_{m}) below, work out b_{n} and also the upper bound of the set {b_{1}, b_{2}, b_{3} … }, when it exists.

a) x_{1} = 1, x_{2} = 2, x_{3} = 3 …, and in general x_{n} = n.

b) x_{1} = 1, x_{2} = 1/2, x_{3} = 1/3 …, and in general x_{n} = 1/n.

c) x_{1} = 1, x_{2} = 2, x_{3} = 1, x_{4} = 2, x_{5} = 1 and so on, alternating between 1 and 2.

d) x_{n} is the sequence defined by x_{n} = 1 if there is a solution to a^{n} + b^{n} = c^{n} in positive integers a, b, c, and x_{n} = 2 otherwise.