THE CHINESE UNIVERSITY OF HONG KONG
Department of Mathematics
MATH 2055 Tutorial 7 (Nov 4 )
Ng Wing Kit
1
1. Let fn (x) = (x + n )2 . Show that fn is not uniformly convergent R.
Solution:
1
Set f (x) = lim fn (x) = lim (x + n )2 = x2
n
n
we need to show t
THE CHINESE UNIVERSITY OF HONG KONG
Department of Mathematics
MATH 2055 Tutorial 6 (Oct 28 )
Ng Wing Kit
1. True or False.
(a) Let f (x) =
lim | 1 |
n n
|x| if x = 0
1 if x = 0
=0=1
lim f (x) does not exist.
n0
Solution: False
By denition of limit of fun
THE CHINESE UNIVERSITY OF HONG KONG
Department of Mathematics
MATH 2055 Tutorial 4 (Oct 7 )
Ng Wing Kit
1. Write down the negations of the following statements.
(a) > 0, N such that n > N , |xn x| <
Solution: > 0, such that N0 , n > N0 , |xn x| >
(b) N ,
THE CHINESE UNIVERSITY OF HONG KONG
Department of Mathematics
MATH 2055 Tutorial 5 (Oct 21 )
Ng Wing Kit
1. True or False.
(a) cfw_xn converges all subsequence cfw_xnk of cfw_xn has a convergent subsequence cfw_xnkl
Solution: False
xn = (1)n is counte
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MATH 2055
Suggested Solution to homework 3
Q2 I denotes the n-th term of each sequences by an
(a) bounded above by sup an = 3, bounded below by inf an = 2, monotone increasing, lim an = 3
nN
n
nN
(b) bounded above by sup an = 2, bounded below by inf an =
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MATH 2055
Suggested Solution to homework 2
Q4 Suppose xn converge to x,
> 0, N , such that n > N , |xn x| <
|xn | |x| |xn x|
<
lim |xn | = |x|
n
Converse is not true. Pick x2m = 1 and x2m+1 = 1 for all natural number m
then (xn ) is divergent while (|xn
THE CHINESE UNIVERSITY OF HONG KONG
Department of Mathematics
MATH 2055 Tutorial 2 (Sep 23 )
Ng Wing Kit
2n
n n!
1. Prove lim
= 0.
2
integer m > 2, m < 1
( to avoid abusing index, we use another index)
> 0,
n > maxcfw_ 4 , 3,
2n
2n
| 0| =
n!
n!
2
2 2 2
THE CHINESE UNIVERSITY OF HONG KONG
Department of Mathematics
MATH 2055 Tutorial 3 (Sep 30 )
Ng Wing Kit
1. Show that if A and B are bounded subests of R . Then A B is a bounded set.
Show that sup(A B) = supcfw_sup A, sup B.
2. Let X and Y be nonempty set
THE CHINESE UNIVERSITY OF HONG KONG
Department of Mathematics
MATH 2055 Tutorial 1 (Sep 16 )
Ng Wing Kit
1. Prove lim ( n + 1 n) = 0
n
> 0,
n > 2 ,
1
n
<
n+1+ n
| n + 1 n 0| = ( n + 1 n)(
)
n+1+ n
1
=
n+1+ n
1
<
n
<
lim ( n + 1 n) = 0
n
2. Prove that