MATH 2122A
Fall 2014
Problem Set #1
Tutorials of October 24 & 26
Note 1: In the following, L:a.b means Lrusson problem a.b while T:x.y.z means Trench
a
problem z in section x.y.
(1) Let s1 = 2 and set sn+1 =
(a) sn < 2 ;
s+
sn . Show
(b) (sn ) converges.
Wednesday November 5th, 2014
MATH 2122A
Department of Mathematics
University of Western Ontario
Makeup Exam v. A
Solutions
(1) (5 marks) State the Axiom of Completeness for a eld F . Every nonempty subset A F
which is bounded above has a least upper bound
MATH 2122A
Fall 2014
Homework 2
Solutions
Note 1: In the following, L:a.b means Lrusson problem a.b while T:x.y.z means Trench
a
problem z in section x.y.
(1) L:2.7 : (i) sup(A) + c is an upper bound for A + c: Given b A + c, we can write b = a + c
for so
MATH 2122A
Fall 2014
Homework 3
Solutions
Perhaps the most surprising thing about mathematics is that
it is so surprising. The rules which we make up at the beginning seem
ordinary and inevitable, but it is impossible to foresee their consequences.
E. C.
MATH 2122a
Fall 2014
Homework 5
Solutions
An expert is a person who has made all the mistakes
which can be made, in a very narrow eld.
Niels Bohr
(1) Use the denition of continuity to show that
(a) f (x) = 2x + 4 is continuous at x = 2; Let > 0 be given.
MATH 2122A
Fall 2014
Homework 4
Solutions
To speak algebraically, Mr. M. is execrable,
but Mr. G. is (x + 1)- ecrable.
Edgar Allen Poe
(1) (a) Prove that if
an converges absolutely, then
a2 also converges absolutely. Ben
cause
|an | converges, the sequenc
MATH 2122A
Fall 2014
Homework 1
Solutions
(1) Using only the axioms A1A5 and the examples seen in class, show that in every commutative eld
(a) the neutral element 0 has no multiplicative inverse;
If x is a multiplicative inverse of 0, then 0 x = 1, contr
MATH 2122A
Solutions
Department of Mathematics
University of Western Ontario
Final Examination
Solutions
In mathematics you dont understand things.
You just get used to them.
J. von Neumann
(1) (5 marks) State the Heine-Borel theorem. A subset of R is com
Solutions Midterm
MATH 2122a, Fall 2014
Department of Mathematics
University of Western Ontario
Midterm
Solutions
(1) (5 marks) Prove that x2 = 6 has no solution in Q. (Proof by contradiction) Suppose (p/q)
is a reduced positive fraction whose square is 6