McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2014
Assignment 6: Solutions
1. Let (xn ) be a sequence. A point x R is called an accumulation point of (xn ) if there exists a
subsequence (xnk ) of (xn ) with lim (xnk
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2014
Assignment 1: Solutions
1. Let f : D E be a function and let A D, B E. Prove the following:
(a) f (f 1 (B) B.
(b) If f is surjective then f (f 1 (B) = B.
(c) f 1 (f
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2014
Assignment 9: Solutions
1. Evaluate the following limits or show that they dont exist. Make sure to only use results that
have been covered in class.
x
xx
x+1
(a) li
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2014
Assignment 3: Solutions
1. Let A and B be two nonempty subsets of R. Prove that A B is bounded above if and only if
both A and B are bounded above. In this case, pro
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2014
Assignment 2: Solutions
1. Let (G, +) be a group. Prove directly from the group axioms that for all a, b, c G we have
a+c=b+c
a=b
c+a=c+b
a=b
This is the so called c
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2014
Assignment 4
You should carefully work out all problems. However, you only have to hand in solutions
to problems 3 and 6.
This assignment is due Tuesday, October 7,
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2014
Assignment 3
You should carefully work out all problems. However, you only have to hand in solutions
to problems 4 and 5.
This assignment is due Tuesday, September 3
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2014
Assignment 2
You should carefully work out all problems. However, you only have to hand in solutions
to problems 3 and 5.
This assignment is due Tuesday, September 2
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2014
Assignment 4: Solutions
1
1. Dene a sequence (xn ) recursively by x1 = 0, x2 = 1, xn+2 = (xn+1 + xn ).
2
(a) Prove by induction that x2k1 < x2k for all k N.
By part
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2014
Assignment 5: 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 2014
Assignment 7
You should carefully work out all problems. However, you only have to hand in solutions to
problems 1 and 5.
This assignment is due Tuesday, October 28,
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2014
Assignment 9
You should carefully work out all problems. However, you only have to hand in solutions to
problems 4, 5(a) and 5(b)(ii).
This assignment is due Thursda
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2014
Assignment 8: Solutions
1. Using the - denition of the limit of a function, prove that
x
a
=
for all a R, a = 1.
xa 1 + x
1+a
x
does not exist.
(b) lim
x1 1 + x
(a)
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2014
Assignment 10: Solutions
1. Let I be a closed and bounded interval and let f : I R be a continuous function such that
f (x) > 0 for all x I. Prove that there exists
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2014
Assignment 10
This is a practice assignment and it will not be marked or collected. However, you
should carefully work out all problems.
1. Let I be a closed and bou
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2014
Assignment 8
You should carefully work out all problems. However, you only have to hand in solutions to
problems 1(a) and 7.
This assignment is due Tuesday, November
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2014
Assignment 7: Solutions
1. Let x1 R \ cfw_0 and let
xn+1 = xn +
1
xn
n N
(a) Prove that lim (xn ) = + if x1 > 0.
(b) Prove that lim (xn ) = if x1 < 0.
Solution:We sh
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2014
Assignment 6
You should carefully work out all problems. However, you only have to hand in solutions
to problems 3(b) and 4.
This assignment is due Tuesday, October
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2015
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 Wednesday, September
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2015
Assignment 2
You should carefully work out all problems. However, you only have to hand in solutions
to problems 2(a), 2(b) and 6.
This assignment is due Wednesday,
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2014
Assignment 8: Solutions
1. Using the - definition of the limit of a function, prove that
x
a
=
for all a R, a 6= 1.
xa 1 + x
1+a
x
does not exist.
(b) lim
x1 1 + x
(
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2016
Midterm 2: Solutions
1. (16 marks) Determine for each of the following sequences whether it converges or diverges.
Determine the limits of the sequences that converg
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2015
Final Exam Solutions
(1)n n
1. (a) (7 marks) Use the definition of the limit of a sequence to show that lim
= 0.
n2 + 1
(b) (8 marks) Determine the following limits;
McGill University
Department of Mathematics and Statistics
MATH 242 Analysis 1, Fall 2016
Midterm 1: Solutions
1. (10 marks) Use induction to prove that 2n1 n for all n N.
Solution:
Base case n = 1: 211 = 20 = 1 1. The statement thus holds for n = 1.
Indu
MATH 242
Final Exam
Page 1 of 1
December 15, 2015
All answers need to be fully justified! Show all your work!
(1)n n
1. (a) (7 marks) Use the definition of the limit of a sequence to show that lim
= 0.
n2 + 1
(b) (8 marks) Determine the following limits;