LECTURE 18: SUMMARY
We began this lecture by proving the following result, which was the final missing step in our proof
of the Cauchy Criterion:
Theorem (Monotone Subsequence Theorem). Every sequence has a monotone subsequence.
Proof. Let (an ) denote a
MAT B43: REMARKS ON FINAL EXAM
The final exam is cumulative, i.e. you should be comfortable with all material taught in the course.
Unless otherwise specified, you may refer to any theorem proved in lecture, without re-proving it
on the exam. However, que
LECTURE 19: SUMMARY
Recall from last time the following definition.
Definition (Metric). Given a nonempty set X, we say that d : X X R is a distance (or metric)
on X iff it satisfies the following three properties:
(1) d(x, y) = 0 iff x = y;
(2) d(x, y) =
LECTURE 14: SUMMARY
We began with the following important result:
Theorem 1 (Monotone Convergence Theorem). Suppose (an ) is a monotone sequence. Then (an )
converges iff (an ) is bounded.
Before proving this, we need to define some of the words appearing
UTSC
MATB43H3
MIDTERM EXAM
STUDY GUIDE
find more resources at oneclass.com
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LECTURE 15: SUMMARY
Today we continued exploring infinite series, in particular proving two important convergence
tests: the root test and the ratio test.
Theorem 1 (Root Test). Suppose (an ) is a sequence of non-negative real numbers, and that
exists.
:
LECTURE 20: SUMMARY
We started by proving the Cauchy-Schwarz inequality:
!1/2
X
X
ai b i
a2i
i
!1/2
X
i
b2i
i
for all ai , bi . Actually, we proved a rather stronger result:
Theorem 1 (Holders inequality). Given p, q > 0 such that 1/p + 1/q = 1. Then
!1/
LECTURE 22: SUMMARY
Today we started exploring point-set topology, the final topic of the term. To motivate the subject,
we started by considering the following question. Suppose f : R R is a continuous function.
What does it do to subsets of R? For examp
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MATB43S
Assignment 2 (2 pages)
Winter 2017
Work on the course material and problems below. Also be sure you read the Notes below.
Tutorial Quiz 2 is based on this assignme
LECTURES 16 AND 17: SUMMARY
In these two lectures, we proved a number of fundamental results about convergence of sequences
and series. We started with an easy observation: that if a sequence converges to some number, then
the terms of the sequence are ev
U NIVERSITY OF T ORONTO S CARBOROUGH
MATB43 : Introduction to Analysis
FINAL EXAMINATION
April 25, 2013
Duration 3 hours
Aids: none
KEY
NAME (PRINT):
Last/Surname
First/Given Name (and nickname)
STUDENT NO:
Qn. # Value Score
1
18
2
12
3
10
4
30
5
30
Total
LECTURE 23: SUMMARY
Last time, we discussed the notion of open sets in a general metric space. We proved that neighbourhoods are open, and that arbitrary unions of open sets are open. What about intersections?
After a bit of thought, we saw that the inter
LECTURE 21: SUMMARY
We continued our discussion of examples of metric spaces. First, we corrected our definition
of the British Rail metric. Recall that we had defined the distance between x and y in Rn as
d(x, y) := |x| + |y|; in other words, to travel f
LECTURE 24: SUMMARY
Last time, we discussed the notion of a limit point: in a metric space (X, d), a point ` X is a
limit point of S X iff ` is the limit of some sequence (of distinct elements) in S. Further, we
showed that this holds iff every neighbourh
LECTURE 13: SUMMARY
There was some concern about the proof from last lecture that any two cardinals are comparable;
since R and N are both well-ordered, cant we use the same technique as in that theorem to prove
that R , N? We discussed why this isnt the
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MATB43S
Assignment 4 (2 pages)
Winter 2017
Work on the material and problems below. Also be sure you read the Notes below on Page 2.
Study for this Assignmnent:
1. Topolog
THE CANTOR-SCHRODER-BERNSTEIN
THEOREM
LEO GOLDMAKHER
A BSTRACT. We give a proof of the Cantor-Schroder-Bernstein theorem: if A injects into B and
B injects into A, then there is a bijection between A and B. This seemingly obvious statement is
surprisingl
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MATB43S
Assignment 1 (2 pages)
Winter 2017
Work on the course material and problems below. Also be sure you read the Notes below.
Tutorial Quiz 1 is based on this assignme
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MATB43S
Assignment 3 (2 pages)
Winter 2017
Work on the course material and problems below. Also be sure you read the Notes below.
Study for this Assignmnent:
1. Innite Ser
LECTURE 12: SUMMARY
The goal of todays lecture was to prove comparability of infinite cardinals: given any two infinite
sets, one of them must inject into the other. Before we can prove this, we need to discuss a bizarre
result called the Well-Ordering Th
APM462H1S, Winter 2016.
Homework 2
(1) Which of the following functions is convex?
The function f (x, y) = xy on the set = cfw_(x, y) E 2 : x 0, y 0.
1
The function f (x, y) = xy
on the set = cfw_(x, y) E 2 : x > 0, y > 0.
The function f (x, y) = log(x
Should human eat meat
Human is heterotrophs organism which mean we need to gain nutrient from other organic
organisms. Meat contains vitamin 12 which act an important character in the normal brain
function and the formation of the red blood cell. It only
v E by > v @E = [r1, r2, , rn].
1
we define the coordinate vector of v relative to E , denoted > v @E or
Definition 3.2:
Let E = (b1, b2, , bn) be an ordered basis for a finite
dimensional vector space V . For v V , with unique representation
v = r1 b1 +
1 .
Addition:
(a + bi) r (c + di) = (a r b) + (c r d)i
1
Geometrically, a complex number can be viewed either as a point or a
vector in the xy-plane, where the x-axis is the real axis and the y-axis is
the imaginary axis.
= cfw_a + bi | a, b
The set of
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
2016 Fall
MATB24H3F
Assignment #4
You are expected to work on this assignment prior to your tutorial in the week of October
3, 2016. You may ask questions about this assignme
You are expected to work on this assignment prior to your tutorial in the week of
October 31, 2016. You may ask questions about this assignment in that tutorial.
In your tutorial in the week of November 7 you will be asked to write a quiz
based on this as
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
2016 Fall
MATB24H3F
Assignment #5
You are expected to work on this assignment prior to your tutorial in the week of
October 17, 2016. You may ask questions about this assignm