Math 312, Intro. to Real Analysis: Midterm Exam #1

Stephen G. Simpson

Friday, February 13, 2009

1. True or False (3 points each) (a) Every ordered field has the Archimedean property.

(b) The ordered field axioms imply |a − b| ≤ |a| + |b| for all a, b.

(c) If lim a_{n} = −∞ then lim sup a_{n} = −∞.

(d) For any sequence of real numbers, the lim inf and the lim sup always exist and furthermore the lim inf is always ≤ the lim sup.

(e) The equation 3x^{3} + 2x^{2} + 3x + 2 = 0 has a rational solution.

(f) ^{3} \sqrt{216} is an irrational number.

(g) The limit of a convergent sequence of negative numbers is negative.

(h) The limit of a convergent sequence of rational numbers is rational.

(i) Every interval contains at least three rational numbers.

(j) Every bounded sequence of real numbers is convergent.

(k) Every convergent sequence of real numbers is bounded.

(l) Every monotone sequence of real numbers is convergent.

(m) If (a_{n}) is a monotone sequence of real numbers, then lim a_{n} exists

and belongs to the interval (−∞, ∞).

2. (7 points each)

(a) Give an example of a sequence of real numbers such that

−∞ < inf a_{n} < lim a_{n} < sup a_{n} < ∞.

(b) Give an example of a sequence of real numbers such that

lim sup a_{n}, lim inf a_{n}, sup a_{n}, inf a_{n}

are four distinct real numbers.

(c) Give an example of a sequence of real numbers such that

lim inf a_{n} = −∞ and lim sup a_{n} = \sqrt{2}.

3. (8 points) It can be shown that ^{3} \sqrt{1 + \sqrt{5}} is an algebraic number, i.e., it is a solution of some polynomial equation with integer coefficients. Find such an equation.

4. (8 points) Find all candidates for rational solutions of the equation

2x^{2} − ax + 5 = 0

where a is an unspecified integer.

5. (12 points) Use algebra plus limit laws to calculate

lim \frac{\sqrt{2n^{2} + 5n}}{n + 4}.

6. (12 points) It can be shown that

lim \frac{^{3} \sqrt{n} − 5001}{^{3} \sqrt{n} − 1001} = 1.

Given ϵ > 0, find an N so large that

| \frac{^{3} \sqrt{n} − 5001}{^{3} \sqrt{n} − 1001} − 1| < ϵ

holds for all n > N.