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(MATH202)[2007](f)final~=fxszzkdm^_18478.pdf
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Math 202 (Introduction to Real Analysis) Fall 2007
Final Examination
Directions: Thisisa closedbook exam. Works (including scratchworks)mustbe shown legibly to receive credits. Answers alone are worth very little!!! Notations: R denotes the set of all real numbers. Part I (Concrete Problems)
1. Let x1,x2,x3,... be a sequence of real numbers suchthat
x1 .2

xn+1 = for n =1,2,3,....
10+ xn
(a)
(11 marks) If x1 = .7, then prove that x1,x2,x3,... converges and .nd its limit.

(b)
(11 marks) If x1 =26, then prove that x1,x2,x3,... converges and .nd its limit.


2. (11 marks)For n=1,2,3,..., let


4n2 . nn.1
yn =+ .
2n2 +nn
Prove that lim yn =3bychecking the de.nitionof limitofa sequence only.

n
(Do not use computation formulas, sandwichtheorem or lHopitals rule! Otherwise, you will get zero mark for this problem.)
Part II (Abstract Problems)
3. Let A and B be nonemptysubsets of R. Both A and B arebounded above. Let C =(A\B)(B \A).
(a)
(3 marks) Give an example of such sets A and B so that C is nonempty and sup C .max{sup A,sup B}.

=

(b)
(6 marks) If C is nonemptyand supC .max{sup A,sup B}, then prove that

= sup(AB)=max{sup A,sup B}.

(c)
(6 marks) If C is nonemptyand supA .sup B, then prove that


= sup C = max{sup A,sup B}.
4. (a) (3 marks) State the de.nition of a sequence x1,x2,x3,... of real numbers converg-ing to a real number L.
a
n
(b) (15 marks) Let a1,a2,a3,... bepositivenumbers suchthat lim =0.
n aa
n+1 +n+2
Prove that a1,a2,a3,... cannotbebounded above.
CEnd ofPaperC