Analysis 1 (MATS2101)

Mathematics and Computer Science - MCS

Semester: First Semester

Level: 200

Year: 2016

Answers (200 )
REPUBLIC O F CA M EROON
Peace-Work-Fatherland
MINIS TRY OF HIGHER
EDUCATION
REPUBLIQUE DU CAMERROUN
Paix-Travaille-Patrie
MLNISTERE DE L'ENSEIGNEIMIENT
SUPERIEURE
THE UNIVERSITY OF BAMENDA UNIVERSITE DE BAM.ENDA
FACULTY OF SCIENCE
P.O Box 39 BA.MBILI
TEL: (+23) 22 81 63 54
FA C U LTE D E S C IEN CE
TUTORIAL SHEET 3 2015/2016
DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE
COURSE CODE: .MATS2101 SHU F. CHE
Answer all questions.
1.
Let{x
n
} be a sequence of real numbers which converges to a
real
number x
0
and
m be an integer. Prove that the sequence
converges to

.
2.
Prove that the sequence {x
n
} where x
n
= (-1)
n
is not convergent.
3.
Let ɛ > 0 be given, ɛ . Find n
o
in terms of ɛ
such that for all n n
o
, |



-
< ɛ and |



-
< ɛ.
4.
Prove that if {x
n
} is a sequence of real numbers and x is a real number, and {x
n
}
converges to x then {|x
n
|} converges to |x|.
5.
Give an example of a sequence {x
n
} such that {|x
n
|} converges but {x
n
} does not
converge.
6.
Let
x
Show that there exists a sequence {x
n
} of rational numbers which
converges to x and
there exists a sequence {y
n
} of rational numbers which
converges to x
7.
Evaluate each of the following limits:
(i) 


(ii) 




(iii) 


8. Give examples of sequences {x
n
} and {y
n
} such that
(i)) {x
n
} and {y
n
} do not converge but {x
n
+ y
n
} converges
(ii) {x
n
} and {y
n
} do not converge but {x
n
y
n
} converges
(ii) {x
n
} and {y
n
} do not converge but
converges.
9. Show that if {x
n
} is a bounded sequence in and {y
n
} converges to 0, then the
sequence {x
n
y
n
} converges to 0.
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10. Give a justified example of a bounded sequence of real numbers that does not
converge.
11. (i) Find a Cauchy sequence of real numbers contained in (0 1) that does not
converge to a point in (0 1)
(ii) Prove that every Monotone sequence of real numbers •which is bounded
converges.
12. Prove that 


= + ∞ and 

! = ∞
13. Complete the following statement: The real numbers x is not the limit of the
sequence {x
n
} of real numbers if …….
14. Prove that the limit of a convergent sequence of real numbers is unique.
15. Prove that the sequence {x
n
} of real numbers with x
n
= n for all n does not
converge.
16. Find n
0
such that for aln n n
0
, |x
n
| < 0.0000000001 and |x
n
| < 0.0000002225
simultaneously, where
xn "#
and y
n
=
for all n $
17. Let {x
n
} and {y
n
} be sequences of real numbers and assume that ({x
n
}{y
n
}) %
where the relation % is as in the definition of real numbers. Show that if {x
n
} is a
cauchy sequence then so is {y
n
} and if {x
n
} converges to a real number x. then so
does {y
n
}
18_ Prove that the set {n: n &} is neither bounded from above nor from below.
19.
Prove that addition and multiplication in is commutative and associative.
20.
Prove that the addition and multiplication in extends the same operations
'.
21.
Show that every constant sequence of real numbers, as (i) bounded, (ii) a Cauchy
sequence (iii) a convergent sequence.
22. Prove that [{0}1 is the additive identity in .
23. Prove that [{–x
n
}]=-[{–x
n
}] where {x
n
} a Cauchy sequence of rational numbers.
24. Prove that every element of has an additive inverse.
12. Prove that the real numbers form a commutative ring.
25. Show that the canonical embedding E : ( , x( )(x):=[{x}]
26. Prove that if x and
y
are positive real numbers then so are x + y, xy and (-x)(-y)
27. Prove that if x and y are positive real numbers and y is a negative real number, then
xy is negative
28. Show that ' is dense in .
29. Find the supremum and infimum of each of the following sets:
i) {x : 0*x
2
* 2}
ii) {x : 0*x}+ {x : x
2
,2}
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30. Let A and B be bounded subsets of . Show that if A - B, then inf B . inf A . sup
A . sup B
31. Let A and B be bounded subsets of Show that sup A u B = max {sup A, sup
B
}
and inf A / B = min {sup A, sup B}
32. Show that every non-empty finite subset of has a minimum and a maximum.
21. Find the supremum and infimum of each of the following sets:
i)
0
1 $
ii)
1 $
3 3 . L e t X - , f : X 2 , g : X 2 p ro v e th a t
(i) sup{ f (x)+ g (x): x X}. sup{ f (x) :x X} + sup{3x) : x X}.
(ii) If f (x). g (x) for all x
X, then sup{f (x) : x X} . sup: {g( x) X} and
inf{f(x) : x X}.{g(x) : x X}
34. Prove each of each of the following statements:
(i) 
2
= 0
(ii) 
2
4

=
(iii) 
2
5

= 0
34. Show that a sequence which is monotone decreasing and converges does so too its
infimum.
36. Prove that for a sequence {x
n
} and a real number x,
(i)
if {x
n
} converges to x, then {|x
n
|} converge to |x|
(ii)
{x
n
} converges to 0 if and only if {|x
n
|} converges to 0.
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