Mathematical Analysis I
Exercises
Lectures for the course Mathematical Analysis I in the winter semester of 2015 are given by Hans Raj Tiwary.
Exercises are intended to supplement the course material with problems
for practice. Weekly exercises are not graded, but students will
present their solutions in the tutorial class the week after receiving
the exercise sheet. Performance on exercises will not directly affect
your grade for the main course. However, if you intend to be graded for
the main course, you require a "pass" for the exercises. A pass/fail
grade for the exercises will be decided based on regular attendence,
participation in presentation of solutions to exercises, and one or two tests
during the semester.
Exercise classes are held at 14:00 on Thursdays (excepting holidays) in
Room S11 in the MFF building at Mala Strana - they immediately follow
the main lecture (after a five-minute break).
Exercise sheets
8 October
There was a typo in 3(i) (an E instead of an R, for the set of reals), which has been corrected in this online version.
A useful alternative textbook to Stephen Abbott's Understanding Analysis is: Robert Bartle and Donald Sherbert, Introduction to real analysis, 3rd. ed.
Solutions to selected exercises.
15 October
This online version of the exercises contains an update to questions 3
(additional part (iii), used to help answer question 4), and an
elaboration of question 4, which previously did not have adequate
contextual results needed in order to prove the countability of the set
of algebraic numbers.
Solutions to selected exercises.
22 October
There is a typo in question 5, where one ``sup B" should be ``sup A" to
make the inequality read sup A less than or equal to sup B.
Question 5 of this sheet is question 8(i)
on the 15 October sheet, and Question 6(i) is contained in the
like-numbered question 6(i) of the 8 October sheet.
Solutions to selected exercises.
29 October
Solutions to selected exercises.
5 November
For question 3(ii) it may be helpful to first prove the case q=1, then
use the identity (x^q-y^q)/(x-y)=x^{q-1}+x^{q-2}y+...+xy^{q-2}+ y^{q-1}
(summation of a geometric progression).
to prove the case p=1. Then combine these two cases to get the result
for arbitrary p/q.
Question 4(ii) may be found quite difficult at this stage: the
easiest approach is probably that which is outlined in Bartle &
Sherbert exercises for section 3.4, number 6, with the result - which
you may assume for the purpose of doing this question - that the
sequence ( (1+1/n)^n ) is bounded above by 3. (This sequence converges
to the constant e.)
Solutions to selected exercises.
12 November
On 19 November a mid-term test given on topics covered hitherto.
Solutions to selected exercises.
26 November
Solution to question 4.
3 December
Solutions to selected exercises.
10 December
Solutions to selected exercises.
17 December
Solution to (some of) question 4.
7 January
Solutions.