1. Let (X, d) be a metric space.
(a) If K1 , ., Kn are compact, then so is K1 Kn .
Proof. Let U be a cover of K = n Ki . Then the U cover each Ki , and so for there are
i,1 , ., i,ki so that Ki j =1 Ui,j . Hence K = i=1 Ki i=1 j =
Math 424 B / 574 B
Some notes on compactness
This short write-up is meant to supplement the discussion on compactness in Chapter 2 of
our textbook. Specifically, the main result below (Theorem 3) is kind of parallel to Theorem
2.41. I will exp
Weeks 8 and 9
November 25, 2013
1 Continuity of functions
Denition 1.1. Let X and Y be metric spaces, f : X Y . We say limxp f (x) = q
or f (x) q as x p for some p X if for all > 0 there is > 0 so that whenever
0 < dX (x, p) < , we have f (x) B (q, ).
October 5, 2013
Let Q denote the rational numbers, i.e.
: p, q Z, q = 0 .
This is an example of a eld, and is natural in the sense that they represent portions
(a half, four thirds, and so on). Of course, there
For a sequence of real numbers an , we will write
an = ap + ap+1 + + aq .
an converges if the limit limN n=1 an converges, and we denote its
limit an . If the sum does not converge, we say an diverges.
Weeks 5 and 6
November 3, 2013
Denition 1.1. Let (X, d) be a metric space and cfw_xn X a sequence of points
in X . We say xn converges to x X if, for every > 0 there is N > 0 such that
n N implies d(x, xn ) < , or in other words, xn B (x,
November 13, 2013
1. Determine whether each of the following series converges or diverges.
n=1 n10 +3n2 +5
n=1 ( n + 1
Proof. (a) This converges by the root test since lim sup
1. For sets A1 , ., An , dene their Cartesian product to be the set of ordered n-tuples
A1 A2 An = cfw_(a1 , ., an ) : ai Ai for every i = 1, 2, ., n.
If A1 = A2 = = An = A, we simply write An for this set.
(a) For d N, show that Nd is countabl
1. Prove that there is no universe, that is, there is no set A containing all possible objects. Hint: to do this, let
A be any set and consider the set
B := cfw_x A : x x,
that is, the sets of things in A that are not subsets of themselves. Sho
1. In each problem below you are given a metric space (X, d) and E X . Show that E is open. (Here, | |
denotes the Euclidean metric)
(a) (X, d) = (Rn+1 , | |), dene a map : Rn+1 Rn by setting (x1 , ., xn+1 ) = (x1 , ., xn ), and
dene E = cfw_x
Math 424/574B, Fall 2013
October 23, 2013
424/574B, Fall 2013
1. Please do not begin the exam until I give the word.
2. When I do give the word, do these rst:
(a) Write your name on the front of the exam.
1. Consider the sequence dened inductively by x1 = 1 and xn+1 =
that xn = 1+2 5 .
(a) Show that = + 1.
Proof. If we solve 2 = + 1 by the quadratic formula,
xn + 1. In this problem, we will prove
is one of the solutions.
(b) Assume that x