Homework 2
Assignment: Read pages 24 to 36 in chapter 2 of Rudin. Whenever the
phrase “metric space” is used, think primarily of the real numbers and absolute value (in fact, whenever you encounter d(p, q) think |p − q|.)
Do problems 6, 9, 10, 11 on pages 43 to 46. written up very neatly. On
number 10, ignore the question about “compactness” for now. Make sure you
know the definition of finite, countable, uncountable, metric, neighborhood,
limit point, isolated point, open, closed, interior, complement, bounded and
dense.
Also do the problems below.
1. The Cartesian product of two sets A and B is the set of ordered pairs
A × B = {(a, b); a ∈ A, b ∈ B}.
(a) Suppose A and B are both countably infinite. Show that A × B
is countably infinite. Hint: number the elements of A as a1 , a2 , . . .
and similarly for B. Now look at the diagram for Theorem 2.12
on page 29 for inspiration. Or, you can even use Theorem 2.12
directly to prove this.
(b) Use part (a) to show that any Cartesian product
A1 × A2 × · · · × Am = {(x1 , x2 , . . . , xm ); xk ∈ Ak }
is countably infinite if each Ai is countably infinite.
2. A real number α is said to be algebraic if α is the root of some polynomial
an xn + an−1 xn−1 + · · · + a1 x + a0 = 0
where the ak are integers.
√
√
√ √
(a) Show that 2, 5, and 2 + 5 are all algebraic.
(b) For any fixed n ∈ N, let An denote the set of algebraic numbers
obtained as roots of polynomials of degree n. Show that An is
countable (recall that an nth degree polynomial has at most n
real roots.) Hint: The conclusion of the previous problem part
(b) may be useful.
1
(c) Use part (b) of this problem to show that the set of algebraic
numbers is countable.
(d) A real number is said to be transcendental if it is not algebraic.
Show that transcendental numbers exist.
2