McGill University
Department of Mathematics and Statistics
MATH 254, Fall 2015
Assignment 3: Solutions
1. Let x be a real number. Show that, for every > 0, there exist two rational numbers q and q
such that q < x < q and |q q | < .
Solution:
We have seen

MATH 254 Hon. Analysis I
Prof. V. Jaki
sc
Solutions for Assignment II.
1. Let A R, B R be two sets bounded from above. The sum of A and B is the set
A + B = cfw_a + b : a A, b B,
Prove that A + B is bounded from above and that
sup(A + B) = sup A + sup B.

McGill University
Department of Mathematics and Statistics
MATH 254 Analysis 1, Fall 2015
Assignment 6: Solutions
1. Let (xn ) be a sequence such that xn > 0 for n N. Set
yn =
1
x1
+
1
x2
n
+ +
n N.
,
1
xn
(a) Suppose that (xn ) is a convergent sequence.

McGill University
Department of Mathematics and Statistics
MATH 254 Analysis 1, Fall 2015
Assignment 5: Solutions
1. Let (yn ) be an unbounded sequence of positive numbers satisfying yn+1 > yn for all n N. Let
(xn ) be another sequence and suppose that th

MATH 254 Hon. Analysis I
Prof. V. Jaki
sc
Solutions for Assignment IV
by Renaud Raqupas
e
partly based on Axel Hundemers solutions
1. Dene a sequence (xn ) recursively by x1 = 0, x2 = 1, xn+2 = 1 (xn+1 + xn ).
2
(a) Prove by induction that x2k1 < x2k for

McGill University
Department of Mathematics and Statistics
MATH 254 Analysis 1, Fall 2015
Assignment 1: Solutions
Some solutions will be only sketched and your are expected to ll in the details.
1. Conjecture a formula for the sum
1
1
1
+
+ +
,
13 35
(2n

McGill University
Department of Mathematics and Statistics
MATH 254 Analysis 1, Fall 2015
Assignment 3
You should carefully work out all problems. However, you only have to hand in solutions
to problems 1,2.
This assignment is due Monday, October 26, at t

McGill University
Department of Mathematics and Statistics
MATH 254 Analysis 1, Fall 2015
Assignment 5
You should carefully work out all problems. However, you only have to hand in solutions
to problems 1, 2, 10, and 11
This assignment is due Monday, Nove

McGill University
Department of Mathematics and Statistics
MATH 254 Analysis 1, Fall 2015
Assignment 6
You should carefully work out all problems. However, you only have to hand in solutions
to problems 1 and 2.
This assignment is due Monday, November 16,

McGill University
Department of Mathematics and Statistics
MATH 254 Analysis 1, Fall 2015
Assignment 4
You should carefully work out all problems. However, you only have to hand in solutions
to problems 3 and 5.
This assignment is due Monday, November 2,

McGill University
Department of Mathematics and Statistics
MATH 254 Analysis 1, Fall 2015
Assignment 1
You should carefully work out all problems. However, you only have to hand in solutions
to problems 2, 5.
This assignment is due Monday, September 21, a

McGill University
Department of Mathematics and Statistics
MATH 254 Analysis 1, Fall 2015
Assignment 2
You should carefully work out all problems. However, you only have to hand in solutions
to problems 2,4.
This assignment is due Monday, October 5, at th

Homework 1 Solutions
1.1.4 (a) Prove that A B i A B = A.
Proof. First assume that A B. If x A B, then x A and x B by
denition, so in particular x A. This proves A B A. Now if x A,
then by assumption x B, too, so x A B. This proves A A B.
Together this imp