Analysis 1 (MATS2101)
Mathematics and Computer Science - MCS
Semester: First Semester
Level: 200
Year: 2014
1. Complete the following statement: The real number x is not the limit of the
sequence {x
n
} of real numbers if……………
2. Prove that the limit of a convergent sequence of real numbers is unique.
3. Prove that the sequence {x
n
} of real numbers with x
n
= n for all n does not
converge.
4. Find n
o
such that
for
aln n
o
> no, |{x
n
}| < 0.0000002225 and |y
n
| < 0.0000002225
simultaneously, where x
n
= (-1)n
and y
n
=
for all n .
5. Let {x
n
} and {y
n
} be sequences of real numbers and assume that ({x
n
}, {y
n
})
. 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
}
6. Prove that the set {n : n } is neither bounded from above nor from
below.
7a Prove that addition and multiplication in is commutative and
associative.
7b Prove that the addition and multiplication in extends the same
operations in.
8. Show
that
every constant sequence of real numbers is (i) bounded, (ii) a
cauchy sequence (iii) a convergent sequence.
9. Prove that [{0}] is the additive identity in
10. Prove that [{-x
n
}] = -[{x
n
}], where {x
n
}a cauchy sequence of rational
numbers.
11. Prove that every element of has an additive inverse.
12. Prove that the real numbers form a commutative ring.
13. Show
-
that the Canonical embedding E : , x E , ( x ) : = [{x}],
i s injective.
14. Pr ove that if x and y are pos itiv e real numbers th en s o are x + y ,
xy and ( -x)( -y)
UNIVERSITY OF BAMENDA
FACULTY OF SCIENCE
MATS2101 TUTORIALS
Department: Mathematics and Computer Science Course Instructor: Dr. Shu F. CHE.
February 10, 2014
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15. Prove that if x is a positive real number and y is a negative real number,
then xy is negative.
16. Show th a t is d e n s e in
17. Find the supremum and infimum of each of the following sets
(1) {x : 0
}
(ii) {x : 0 {x :
}
18. Let A and B be bounded submits of . Show that if A B; then inf B
inf A sup A sup B
19. Let A and B be bounded subsets of . Show that sub AB = max{sup A, sup B}
and inf AB = min{sup A, sup B}
20. Show that every non-empty finite subset of has a minimum and a
maximum
21. Find the supremum and infinum of each of the following ;obis
(i) {1 -
: n }
(ii) {n -
: n
22. Let X f : X , y : X . Prove that
(i) sup{ f (x) +g (x) : x X } ! sup{g (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} inf{g(x) : x
X
}.
23. Prove each of each of the following statements:
(i) "#$
%
= 0
(ii) "#$
%
&'(
)'
=
&
)
(iii) "#$
%
*
'
= 0
24. Show that a sequence which is monotone decreasing and converges
does so to its infimum.
25. 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.
26. Show by example that you may have 4 sequence {x
n
} which does not
converge but |{x
n
}|does converge.
27. Find the limits of the following sequences {x
n
}:
(i) x
n
:=
'
(ii) x
n
:= (3 +
'
)
2
(iii) x
n
:=
+
, -
+
, - .
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(iv) x
n
:= n -
+
,
- , (v) x
n
:=
'
')
27. Find examples of sequences {x
n
} and {y
n
}of real numbers which do
not converge but
( i) {x
n
+ y
n
}
converges
(ii) {x
n
y
n
} converges
(iii)
/
0
1
0
converges
28. Let {x
n
} be sequences of real numbers such that {x
n
+ y
n
}
and
{x
n
- y
n
}
converge. Show that {x
n
} and {y
n
} converge
29. Let {x
n
} and {y
n
} be sequences of real numbers such that {x
n
} is bounded
and {y
n
} converges to 0. Show that {x
n
y
n
} converges to 0.
30. Give an example of an unbounded sequence which has a convergent stiles
31. Give an example of tut unbounded sequence which has no convergent
subsequence.
32. Let {x
n
} and {y
n
} be sequences of real numbers such that x
n
y
n
z
n
for
all n. Show that if (x
n
) and {z
n
} converge to a real number x, then so does
{y
n
}.s
33. Let {x
n
} he a sequence of real numbers defined as follows: x
1
:-1 and for
n 2 2 let x
n
: =
(2x
n-1
+ 3). Show that {x
n
} converges.
34. Let. {x
n
} be a sequence of real numbers defined as follows: x
1
: =3 and
for n 2 2 let x
n
: =2(
/
034
).
35. Find a cauchy sequence in |0 1| which does not converge to a point in |0 1|.
36. Let x
n
:=
5
6
67
, n 21. Show that the "#$
%
8
'
9 8 - 0 but {x
n
} is not a
cauchy sequence
37. Let a, b , a < b. Let
x
1
=
a, x
2
= b and x
n
=
/
:3
'/
034
, n 3. Prove
that {x
n
} cauchy sequence.
38. Let r
be such that 0 < r < 1 and {x
n
} be a sequence of real numbers
such that |x
n+2
– x
n+1
| < r|
xn+1
– x
n
| for all n. Show that {x
n
} is a cauchy
sequence.
39. Let x
1
> 0 and x
n+1
= for n 2 1. Using question 38, show that {x
n
}
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is a Cauchy sequence and hence find the limit of the sequence.
42. Prove each of the following statements about a sequence {x
n
} of real numbers
(i) {x
n
}converges to if and only if {x
n
} converges to ∞
( i i ) I f x
n
> 0 for all n then {xn} converges to ∞ if and only if
/
0
converges
to 0
(iii) If x
n
< 0 for all n then {x
n
} converges to - ∞ if an only if |x
n
|
(iv) If {x
n
} is not bounded from above, then it has a subsequence which
converges to ∞
43. Find Iim sup x
n
and lim inf x
n
for each of the following sequences {x
n
}:
(i) x
n
%
7
9.
(ii)
%
/
0
cos(n; (iii) x
n
= (9.
(1 +
) (iv) x
n
(1 +
)sin(n;
44. Prove each of the following statements about sequences {x
n
} and {y
n
} of real
numbers:
(i) If x
n
y
n
for all n, then "#$
%
#<=
"#$
%
#<=>
and "#$
%
"#$
%
>
(ii) "#$
%
#<=
+ "#$
%
#<=>
"#$
%
#<=
- >
(iii) "#$
%
#<=
+ "#$
%
#<=>
2 "#$
%
#<=
- >
(iv) "#$
%
#<=
= "#$
%
9
(v)
45. Show that the series
5
66'
%
67
converges
46. Show that the series
5
6
%
67
converges if and only if |x| = 1
47. Show that if the series
5
%
67
and
5
>
%
67
converge and ? then so does
the series
5
->
%
67
and
5
@
%
67
and in this case
5
->
%
67
=
5
%
67
+
5
>
%
67
and
5
@
%
67
=
5
%
67
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