Math 320 (Winter 2006)
Introduction to Real Analysis II
Midterm Solutions
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Q1.
(a)
Ai Bn Ai Bn
n
Ai Bn .
i=1
On the other hand,
n
Bn
n
Ai Bn
i=1
Ai
i=1
n
Bn
Ai ,
i=1
because
(b)
n
i=1 Ai
is closed.
Ai Ai B
B
First-Order Linear Partial Differential Equations
Let u = u ( x, y ) and consider: a u x + bu y + c ( x, y ) u = g ( x, y ) , where a and b are non-zero constants and
c ( x, y ) and g ( x, y ) are continuous on some open region of the x-y plane.
p q x p
Homework 1
Math 320
Introduction to Analysis II
Due Date: Jan 13, 2017, Friday, 5:00 pm
1. For x R and y R, define
d1 (x, y) = (x y)2 , d2 (x, y) =
p
|x y|, d3 (x, y) = |x2 y 2 |,
|x y|
.
1 + |x y|
Determine, for each of these, whether it is a metric or n
Homework 2
Math 320
Introduction to Analysis II
Due Date: Jan 20, 2017, Friday, 5:00 pm
1. Let K R consist of 0 and the numbers 1/n, for n = 1, 2, 3, . . . Prove that K is compact
directly from the definition (without using the Heine-Borel theorem).
2. Co
Math 320
Introduction to Real Analysis II
Assignment 1 Solutions
Desmond Leung
Report corrections to DesmondL at sfu dot ca
Q1. Clearly, for any positive integer N , there exists nitely many (n + 1)-tuple
integers (a0 , . . . , an ) such that
|a0 | + + |a
Math 320 (Winter 2006)
Introduction to Real Analysis II
Assignment 4 Solutions
Report corrections to DesmondL at sfu dot ca
Q1. By Cauchy-Schwarz inequality,
n=1
an
n
( an )
n=1
2
1
n
an
n=1
n=1
Since
1/n2 converges (p-series with p > 1) and
assumption, t
Math 320 (Winter 2006) Introduction to Real Analysis II Assignment 5 Solutions Report corrections to DesmondL at sfu dot ca Q1. Clearly, f (E) f (E), and so E f -1 (f (E). Since f is continuous, f -1 (f (E) is closed. But by definition, E is the smallest
Math 320 (Winter 2006) Introduction to Real Analysis II Assignment 6 Solutions Report corrections to DesmondL at sfu dot ca Q1. For all partitions P , we have L(P, f, ) = 0. For U (P, f, ), for any partition P = cfw_a = x1 < x2 < < xn = b with xi1 < x0 xi
Math 320 (Winter 2006)
Introduction to Real Analysis II
Assignment 9 Solutions
Report corrections to DesmondL at sfu dot ca
n
m=1 fm (x)
Q1. Let An (x) :=
so that
|An (x)|
for all n and all x in E. Also, for any
|gn (x)| <
> 0 there is N such that
2M
for
Math 320 (Winter 2006)
Introduction to Real Analysis II
Assignment 8 Solutions
Report corrections to DesmondL at sfu dot ca
Q1. Let cfw_fn be a sequence of functions belonging to some set E that
converges to f , with fn bounded by Mn (here, this bound de
First-Order Partial Differential Equations - Simple Case
Propagation of Discontinuities in the Initial Data
Discontinuities in the initial data (or any of its derivatives) are propagated along the characteristics.
0, y < 2
Example 1: u x = u ; I.C. (Ini