Department of Mathematics and Statistics McGill University
MATH 242 Assignment 1 Due in class, Monday September 21
Full credit can be obtained from correct answers to questions totalling 70 points. 1. (10 points) Let f : X Y , A X and B Y , then prove or
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2016
Assignment 4: Solutions
1. Let S := cfw_
1
1
: n, m N. Determine sup S and inf S.
n m
Solution:
For all n, m N we have that
1
1
1
1
1
m
n m
n
Hence 1 is an upper
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2016
Assignment 1
You should carefully work out all problems. However, you only have to hand in solutions
to problems 2 and 5.
This assignment is due Thursday, September
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2016
Assignment 4
You should carefully work out all problems. However, you only have to hand in solutions
to problems 2 and 4.
This assignment is due Thursday, October 13
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2016
Assignment 5: Solutions
1. Use the definition of the limit of a sequence to show that:
2
n
1
n 1
=0
(b) lim
=
(a) lim
2
n+1
2n + 3
2
(c) lim
(1)n n
n2 + 1
=0
Solu
MATH 242 Assignment 1 Solutions
1. TRUE. We have y f (A f 1 (B ) x A f 1 (B ) such that f (x) = y x A such that f (x) B and f (x) = y (x A such that f (x) = y ) and y B y f (A) and y B y f (A) B 2. FALSE. You can start out by arguing as in question 1. y f
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2016
Assignment 6: Solutions
1. If (bn ) is a bounded sequence and lim (an ) = 0, prove that lim (an bn ) = 0.
Solution:
Let M > 0 such that |bn | M for all n N and let >
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2016
Assignment 2: Solutions
1. (a) Let x R be irrational and r Q, r 6= 0, be rational. Prove that x + r and x r are
irrational.
(b) Prove that 6 and 2 + 3 are irrational
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2016
Assignment 7: Solutions
1. Show directly from the definition that the following sequences are Cauchy sequences:
1
1
1
1
1
1
(b) 1 + 2 + 2 + + 2
(a) 1 + + + +
1! 2!
n
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2016
Assignment 3
You should carefully work out all problems. However, you only have to hand in solutions
to problems 1 and 2.
This assignment is due Tuesday, October 4,
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2016
Assignment 8
You should carefully work out all problems. However, you only have to hand in solutions to
problems 3 and 4.
This assignment is due Tuesday, November 15
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2016
Assignment 7
You should carefully work out all problems. However, you only have to hand in solutions
to problems 2 and 4.
This assignment is due Tuesday, November 8,
Department of Mathematics and Statistics McGill University
MATH 242 Assignment 2 Due in class, Wednesday October 7
Full credit can be obtained from correct answers to questions totalling 70.
n n
1. (10 points) Let n N and let a1 , a2 , . . . , an 0. Show
MATH 242 Assignment 2 Solutions
1. Proof by induction on n. In case n = 1 the statement reduces to 1 + a1 1 + a1 which is clearly true. So the induction starts. Now suppose that the statement is true for n we will establish it for n + 1. We are
n n
assumi
Department of Mathematics and Statistics McGill University
MATH 242 Assignment 3 Due in class, Friday October 23
Full credit can be obtained from correct answers to questions totalling 70. 2x3 + a n and verify that xn+1 > 0 3x2 n
1. (10 points) Let a > 0
MATH 242 Assignment 3 Solutions
1.
(i) It is easy to verify that xn+1 xn = x3 a n . 3x2 n (1)
We have x3 +1 a = n = On the other hand (8x3 + a)(x3 a)2 (8x3 + a)(x6 2ax3 + a2 ) 8x9 15ax6 + 6a2 x3 + a3 n n n n n n n = =n = x3 +1 a n 27x6 27x6 27x6 n n n (ii
Department of Mathematics and Statistics McGill University
MATH 242 Assignment 4 Due in class, Friday November 6
Full credit can be obtained from correct answers to questions totalling 70.
2
1. (10 points) Let c R and f : R R be such that lim f (x)
xc
= L
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2016
Assignment 5
You should carefully work out all problems. However, you only have to hand in solutions
to problems 2 and 4(a)+(b).
This assignment is due Thursday, Oct
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2016
Assignment 2
You should carefully work out all problems. However, you only have to hand in solutions
to problems 1 and 6.
This assignment is due Tuesday, September 2
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2016
Assignment 3: Solutions
1. Let a, b R with a 6= b. Use the triangle inequality to prove that there exists an > 0 such
that the -neighborhoods of a and b are disjoint
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2016
Assignment 6
You should carefully work out all problems. However, you only have to hand in solutions
to problems 2 and 3.
This assignment is due Thursday, October 27