Math 131AH Fall 2015: Homework 8, Due 11/20
1. Let (X, d) be a metric space.
(a) Show that X is connected if and only if every continuous function f : X
N is a constant.
(b) Consider a continuous function g : X Y , where Y is another metric
space. Using
Math 131A, Homework 1
Sep. 30th
Xinru Hua
1: If r is rational (r = 0) and x is irrational, prove that r + x and rx is irrational.
Proof. Since r is rational, let r = p , (p, q Z), (p = 0).
q
(a) Assume r + x is rational, then r + x = m , (m, n Z), (n = 0)
infimum and supremum for real
numbers
matte
2013-03-21 20:04:34
Suppose A is a non-empty subset of R. If A is bounded from above, then
the axioms of the real numbers imply that there exists a least upper bound for
A. That is, there exists an m R such that
Math 131AH - Old Midterm Exam, Oct. 26, 2004
Write your answers on your exam (which had 7 pages in 2004). You may remove the
scratch paper of the end of your exam. All questions have equal value.
1. In the natural numbers N = cfw_0, 1, . recall that we ha
Math 131AH Fall 2011: Homework 6, due in class 11/4
1. Let us define the metric space
2
X = cfw_~a = (a1 , a2 , a3 , .) (R)N such that
k=1 |ak | 1.
2 1/2
with the metric d(~a, ~b) := [
.
k=1 |ak bk | ]
(a) Show that (X, d) is a metric space.
(b) Show tha
Some Exercises on Induction, Problem 1 is for homework 1
1. Show the following theorem:
Theorem 0.1 (Unique Factorization Theorem). Every integer n > 1 can be represented as
a product of prime factors in only one way, apart from the order of the factors.
Math 131AH Fall 2015: Homework 3, Due 10/14
1. Let E be a nonempty set. Show that E is infinite if and only if E has the
same cardinality with at least one of its proper subset.
2. Let E be any collection of disjoint intervals in R with positive length.
S
Assignment Five due Wednesday, May 11
29. Suppose that cfw_fn is a sequence of Riemann-integrable functions on [a,
xb] that is
uniformly bounded: |fn (x)| B for all x [a, b] for all n. Let Fn (x) = a fn (t)dt.
Show that cfw_Fn has a subsequence that is
Math 131AH Fall 2015: Homework 2, Due 10/7
1. Let A and B be nonempty subsets of R+ := cfw_x R : x > 0 which are
bounded above. Let us define
C = cfw_xy : x A and y B.
Show that sup C = (sup A)(sup B).
2. Show that irrationals are dense in R: i.e. prove t
Math 131AH Fall 2015: Homework 4, Due 10/21
1. Let E be a non-empty subset of R that is both open and closed. Show
that E = R.
2. Show that any open subset of R is at most countable union of disjoint
open intervals.
3. For the space of polynomials
X := cf
131AH Selected Homework 4 Problems
Here are some solutions or partial solutions. There are multiple ways to do most of these problems.
1. Let E be a non-empty subset of R that is both open and closed. Show that E = R.
Suppose for the sake of contradiction