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Math 341 Homework 1
September 18, 2014
Problem 1.1.17 For a, b R with a < b, nd an explicit bijection of
A := cfw_x : a < x < b onto B := cfw_y : 0 < y < 1.
Proof.
Consider the line through the points (a, 0) and (b, 1),
f (x) =
xa
.
ba
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Math 341 Homework 1
December 3, 2014
Problem 2.4.18. If u > 0 is any real number and x < y, show that there exists a
rational number r such that x < ru < y. (Hence the set cfw_ru : r Q is dense in R.)
Proof.
From the denseness of Q in R
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Math 341 Homework 4
October 10, 2014
3.4.10. Let xn be a bounded sequence and for each n N let sn := supcfw_xk : k n
and S := infcfw_sn . Show that there exists a subsequence of xn that converges to S.
Proof.
Recall that S = infcfw_sn
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Math 341 Homework 1
October 10, 2014
3.2.19 Discuss the convergence of the following sequences, where a, b satisfy 0 < a < 1,
b > 1.
(b) (bn /n2 )
(c) (bn /n!)
(d) (n!/nn ).
Proof.
(b) Let xn = bn /n2 and Ln = xn+1 /xn . Then limn Ln =
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Math 341 Homework 5
October 14, 2014
Problem 1. (a) Let fn be the sequence of Fibonacci with initial values f1 = f2 = 1;
fn
1
and xn be the sequence dened by xn = fn+1 . Show that 1 1+xn 2 by induction.
2
3
(b) Let xn be a contractive s
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Math 341 Homework 7*
November 4, 2014
1. (Alternating Series Test.) Take zn = 1/n2 . Show that
s2n = (z1 z2 ) + (z3 z4 ) + + z2n1 z2n > 0,
Note that if n < n+1, then 1/n > 1/(n+1) and 1/n2 > 1/(n+1)2 . Hence, z2i1 z2i > 0
for each i = 1
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Math 341 Homework 6
October 30, 2014
Problem 9.2.1,2. Establish the convergence or divergence of the series whose nth term
is:
(b)
n
(n+1)(n+2)
(c) n!/nn
Proof.
(b) We will use the comparison test. (Note that the ratio and root test are
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Math 341 Homework 8
November 21, 2014
Problem 1. Let S be a subset of the real numbers. Show that x is a cluster point of S
i there exists a sequence xn S such that limn xn = x.
Proof.
= Let x be a cluster point of S, and let n = 1/n. T
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Math 341 Homework 9
November 29, 2014
Problem 4.2.12. Let f : R R be such that f (x + y) = f (x) + f (y) for all x, y R.
Asuume that limx0 f = L Prove that L = 0, and then prove that f has a limit at each
point c R.
Proof.
Let y = 2x. T
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Math 341 Homework 10
November 21, 2014
Problem 5.3.1. Let I = [a, b] and let f : I R be a continuous funciton such that
f (x) > 0 for each x in I. Prove that there exists an > 0 such that f (x) for all
x I.
Proof.
Using the Max-Min Theo
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Math 341 Homework 13
December 19, 2014
Problem 8.1.9. Show that lim(x2 enx ) = 0 and lim(n2 x2 enx ) = 0 for x R, x 0.
Proof.
(i) If x = 0, then x2 enx = 0 1 = 0 and so limn (x2 enx ) = 0. If x > 0, then using
Lhopitals rule
x2 L h
2x
=
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Math 341 Homework 12
December 11, 2014
Problem 1. Let f be an uniformly continuous function on (a, b). Show that there exists
a unique continuous extension f on [a, b].
Proof.
For the existence of such an extension f , see Theorem 5.4.8
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Math 341 Homework 11
December 5, 2014
Problem 11.3.1. Let f : R R be dened by f (x) = x2 for x R.
(a) Show that the inverse image f 1 (I) of an open interval I := (a, b) is either an open
interval, the union of two open intervals, or em
Midterm Exam, math 341, Fall 2014
Print your last name:
Circle the time of your class:
Instructions:
rst name:
(061) 3:00 pm,
(062) 4:30 pm.
1. This exam contains 12 pages. The last page is left intentionally blank, which you may use as scrap
paper.
2. Th