Loyola College B.Sc. Mathematics April 2009 Real Analysis Question Paper PDF Download



ZA 29


MT 5505 / 5501 – REAL ANALYSIS




Date & Time: 16/04/2009 / 9:00 – 12:00       Dept. No.                                                       Max. : 100 Marks





Answer all questions:                                                                        (10 x 2=20)


  1. Define an order – complete set and give an example of it.
  2. When do you say that two sets are similar?
  3. Define discrete metric space.
  4. Give an example of a perfect set in real numbers.
  5. Define open map and closed map.
  6. When do you say that a function is uniformly continuous?
  7. If a function f is differentiable at c, show that it is continuous.
  8. Define “total variation” of a function f on [a,b].
  9. Give an example of a sequence {an}whose lim inf and lim Sup exist, but the sequence is not convergent.
  10. Give an example of a function which is not Riemann Stieltjes integrable.




Answer any five questions:                                                              (5 x 8=40)


  1. Let a,b be two integers such that (a,b)=d. Show that there exists integers  such that .
  2. Show that the set of all real numbers is uncountable.
  3. Let E be a subset of a metric space X. Show that  is the smallest closed set containing E.
  4. State and prove Heire Borel theorem.
  5. Show that in a metric space every convergent sequence is Cauchy, but not conversely.
  6. Let f, g be differentiable at Show that f  g  is differentiable at c and if is also differentiable at c.
  7. State and prove Lagrange’s mean valve theorem.
  8. Suppose {an}is a real sequence. Show that lim Supan=l if and only if for every ,
  • there exists a positive integer N such that for all and
  • given any positive integer m, there exists an integer such that .









Answer any two questions:                                                                          (2 x 20=40)    


  1. (a) State and prove Minkowski’s inequality.

(b) Show that there is a rational number between any two distinct real numbers.

(c) If show that e is irrational.

  1. (a) If F is a family of open intervals that covers a closed interval [a,b], show that a finite sub family of F also covers [a,b].

(b) Let SÌRn. If every infinite subset of S has an accumulation point in S, show that S is closed and bounded.

  1.  (a) Let (X,d1), (Y, d2) be metric spaces and . Show that f is continuous at if and only if for every sequence in X that converges to  the sequence converges to .

(b) State and prove Bolzano theorem.


  1. (a) State and prove Taylor’s theorem.

(b) Suppose on [a,b]. Show that on [a,b] and



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