Real Analysis Test-01 - RAMANASRI IAS...
Transcript of Real Analysis Test-01 - RAMANASRI IAS...
![Page 1: Real Analysis Test-01 - RAMANASRI IAS INSTITUTEmathematicsoptional.com/wp-content/uploads/2016/06/... · Reputed Institute for IAS, IFoS Exams Page 2 1. (i) Show that there is no](https://reader033.fdocuments.net/reader033/viewer/2022043023/5f3ee0853dfa946c080fddd5/html5/thumbnails/1.jpg)
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
![Page 2: Real Analysis Test-01 - RAMANASRI IAS INSTITUTEmathematicsoptional.com/wp-content/uploads/2016/06/... · Reputed Institute for IAS, IFoS Exams Page 2 1. (i) Show that there is no](https://reader033.fdocuments.net/reader033/viewer/2022043023/5f3ee0853dfa946c080fddd5/html5/thumbnails/2.jpg)
Reputed Institute for IAS, IFoS Exams Page 2
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)
![Page 3: Real Analysis Test-01 - RAMANASRI IAS INSTITUTEmathematicsoptional.com/wp-content/uploads/2016/06/... · Reputed Institute for IAS, IFoS Exams Page 2 1. (i) Show that there is no](https://reader033.fdocuments.net/reader033/viewer/2022043023/5f3ee0853dfa946c080fddd5/html5/thumbnails/3.jpg)
Reputed Institute for IAS, IFoS Exams Page 3
2. Show that the set of real numbers as a complete ordered field (15 Marks)
![Page 4: Real Analysis Test-01 - RAMANASRI IAS INSTITUTEmathematicsoptional.com/wp-content/uploads/2016/06/... · Reputed Institute for IAS, IFoS Exams Page 2 1. (i) Show that there is no](https://reader033.fdocuments.net/reader033/viewer/2022043023/5f3ee0853dfa946c080fddd5/html5/thumbnails/4.jpg)
Reputed Institute for IAS, IFoS Exams Page 4
3. Every subset of a countable set is countable (15 Marks)
![Page 5: Real Analysis Test-01 - RAMANASRI IAS INSTITUTEmathematicsoptional.com/wp-content/uploads/2016/06/... · Reputed Institute for IAS, IFoS Exams Page 2 1. (i) Show that there is no](https://reader033.fdocuments.net/reader033/viewer/2022043023/5f3ee0853dfa946c080fddd5/html5/thumbnails/5.jpg)
Reputed Institute for IAS, IFoS Exams Page 5
![Page 6: Real Analysis Test-01 - RAMANASRI IAS INSTITUTEmathematicsoptional.com/wp-content/uploads/2016/06/... · Reputed Institute for IAS, IFoS Exams Page 2 1. (i) Show that there is no](https://reader033.fdocuments.net/reader033/viewer/2022043023/5f3ee0853dfa946c080fddd5/html5/thumbnails/6.jpg)
Reputed Institute for IAS, IFoS Exams Page 6
4. The countable union of countable sets is countable (10 Marks)
![Page 7: Real Analysis Test-01 - RAMANASRI IAS INSTITUTEmathematicsoptional.com/wp-content/uploads/2016/06/... · Reputed Institute for IAS, IFoS Exams Page 2 1. (i) Show that there is no](https://reader033.fdocuments.net/reader033/viewer/2022043023/5f3ee0853dfa946c080fddd5/html5/thumbnails/7.jpg)
Reputed Institute for IAS, IFoS Exams Page 7
5. The set of rational numbers is countable (15 Marks)
![Page 8: Real Analysis Test-01 - RAMANASRI IAS INSTITUTEmathematicsoptional.com/wp-content/uploads/2016/06/... · Reputed Institute for IAS, IFoS Exams Page 2 1. (i) Show that there is no](https://reader033.fdocuments.net/reader033/viewer/2022043023/5f3ee0853dfa946c080fddd5/html5/thumbnails/8.jpg)
Reputed Institute for IAS, IFoS Exams Page 8
6. Limit of sequence, if it exist, is unique or A sequence cannot converge to more than one
point (10 Marks)
![Page 9: Real Analysis Test-01 - RAMANASRI IAS INSTITUTEmathematicsoptional.com/wp-content/uploads/2016/06/... · Reputed Institute for IAS, IFoS Exams Page 2 1. (i) Show that there is no](https://reader033.fdocuments.net/reader033/viewer/2022043023/5f3ee0853dfa946c080fddd5/html5/thumbnails/9.jpg)
Reputed Institute for IAS, IFoS Exams Page 9
7. Show that every convergent sequence is bounded. What about converse, true or false?
Give an example? (10 Marks)
![Page 10: Real Analysis Test-01 - RAMANASRI IAS INSTITUTEmathematicsoptional.com/wp-content/uploads/2016/06/... · Reputed Institute for IAS, IFoS Exams Page 2 1. (i) Show that there is no](https://reader033.fdocuments.net/reader033/viewer/2022043023/5f3ee0853dfa946c080fddd5/html5/thumbnails/10.jpg)
Reputed Institute for IAS, IFoS Exams Page 10
8. Show that the sequence n
f defined by 1 n
f n n n N is convergent
(10 Marks)
![Page 11: Real Analysis Test-01 - RAMANASRI IAS INSTITUTEmathematicsoptional.com/wp-content/uploads/2016/06/... · Reputed Institute for IAS, IFoS Exams Page 2 1. (i) Show that there is no](https://reader033.fdocuments.net/reader033/viewer/2022043023/5f3ee0853dfa946c080fddd5/html5/thumbnails/11.jpg)
Reputed Institute for IAS, IFoS Exams Page 11
9. Show that 1
lim 1n
nn
(10 Marks)
![Page 12: Real Analysis Test-01 - RAMANASRI IAS INSTITUTEmathematicsoptional.com/wp-content/uploads/2016/06/... · Reputed Institute for IAS, IFoS Exams Page 2 1. (i) Show that there is no](https://reader033.fdocuments.net/reader033/viewer/2022043023/5f3ee0853dfa946c080fddd5/html5/thumbnails/12.jpg)
Reputed Institute for IAS, IFoS Exams Page 12
10. limn
f l limn
f l but the converse is not true (15 Marks)
![Page 13: Real Analysis Test-01 - RAMANASRI IAS INSTITUTEmathematicsoptional.com/wp-content/uploads/2016/06/... · Reputed Institute for IAS, IFoS Exams Page 2 1. (i) Show that there is no](https://reader033.fdocuments.net/reader033/viewer/2022043023/5f3ee0853dfa946c080fddd5/html5/thumbnails/13.jpg)
Reputed Institute for IAS, IFoS Exams Page 13
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)