real analysis test-01 - ramanasri ias...

2016

Real Analysis Test-01

Covered topics: Real number system as an

ordered field with least upper bound

property; Sequences, limit of a sequence,

Cauchy sequence, completeness of real line;

Ramanasri

S H O P N O - 4 2 , 1 S T F L O O R , N E A R R A P I D F L O U R M I L L S , O L D R A J E N D E R N A G A R , N E W D E L H I C O N T A C T : 8 7 5 0 7 0 6 2 6 2 / 6 3 6 3 / 6 4 6 4

Reputed Institute for IAS, IFoS Exams

1. (i) Show that there is no rational number whose square is 2 (4 Marks)

(ii) Show that there is no rational number whose square is 8 (6 Marks)


2. Show that the set of real numbers as a complete ordered field (15 Marks)


3. Every subset of a countable set is countable (15 Marks)


4. The countable union of countable sets is countable (10 Marks)


5. The set of rational numbers is countable (15 Marks)


6. Limit of sequence, if it exist, is unique or A sequence cannot converge to more than one

point (10 Marks)


7. Show that every convergent sequence is bounded. What about converse, true or false?

Give an example? (10 Marks)


8. Show that the sequence n

f defined by 1 n

f n n n N is convergent

(10 Marks)


9. Show that 1

lim 1n

nn

(10 Marks)


10. limn

f l limn

f l but the converse is not true (15 Marks)


11. Show that the sequence n

f where 1 1 1

1 ...3 5 2 1

nf

n

is not a Cauchy sequence. Is

it convergent (10 Marks)

real analysis test-01 - ramanasri ias...

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