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

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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)

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

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

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

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

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

point (10 Marks)

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

Give an example? (10 Marks)

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8. Show that the sequence n

f defined by 1 n

f n n n N is convergent

(10 Marks)

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9. Show that 1

lim 1n

nn

(10 Marks)

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10. limn

f l limn

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

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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)