MATH 520: Foundations of Analysis

Journal

HW 13 (due 5/5, 6pm):

Page 118, #2, #3, #6, #8, #11.
Page 128, #12.
Carefully study example 6 on page 124.
Page 135, #5 a), d).
Page 141, #4, #8.
Page 144, #1 e), #2 a), #9.

HW 12 (due 4/28, 6pm):

Page 103, #4, #5, #7, #8 g), #9.
Page 110, #5 a), b), # 6 b), d),
#10, #12, #14, #16, #29.
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Practice Final

Your final is scheduled We 5/11, 11:50-1:40.
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HW 11 (due 4/21, 6pm):

Page 91, #2 e) (this is an example of the general result: " a continuous function that has limit zero at +/-
infinity is uniformly continuous on the real line"; the proof is done the same way in the general case; # 2 f)
is more of the same... don't do #2 f), rather try to do the proof of the general result slightly tweaking your
proof of #2 e)). #5 ( part b) can be done with the technique for #9 that I showed in class), #6, #7, #8, #10.
In class we used that |sin x|<= |x|. How this is done can be found on page 67, #9 a),
using geometrical ideas - the picture is Fig. 3.2.3..

Page 99, #6, #7 c), e), #9, #11 a), #12 b), #14.

HW 10 (due 4/14, 6 pm):

Page 84, #3, #4, #6, #8,# 9, #12, #13, #17.
#18, ( a) is true. How about b)?), #19, #20 (adapt some example that came up in a recent section...
both a) and b) should be false). #21. Give proofs showing that things are as you claim!
Page 91, #1 c), #2 b), #3 b), #4.

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HW 9 will be due 4/7, 6 pm. We will continue working on HW on Wednesday.
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HW 9 (due 4/5, 6 pm):

Page 70, #1 f), #3 a), b), c), #5 e), f), g), #7, #9.
Page 77, #1 b), i), l), #2, #4, #9, #11 a), c).
#3, #6, #7, #10,

HW 8 (due 3/15, 6 pm):

Page 63, #2 g), j), l), #3, #4, #6, #10.
Page 66, #1, #2, #5, #6, #7, #9.

HW 7 (due 3/10, 6 pm):

Page 55, #3, #4 a), b), c), #8 b), #9, #11, #12.
Page 57, #1 c), d), f), g) #2 b), c), #3, #7, #8.

Your midterm will be on 3/18. Here is a practice midterm.

HW 6 (due 3/1, 6 pm):

Page 46, #1 a), c) ,d); #2, #3, #5, #8 (a) and b) are vey very easy, c) is a lot of work), #10.
#7 a).
Page 50, #5 a), f), #6 b), #7 d), e) f).

Note: About #8:
a) The limit is a trascendental number (It was proved in ??? that this number is actually trascendental,
that is, it is not a root of any polynomial with integer coefficients. This number is of the type called a "Liouville"
number, in honor of Liouville, who proved that trascendental numbers exist). (I picked this exercise not for
the techniques that solving it requires - it's easy - but to get you aquainted with this number).

c) The limit is e (the base of natural logarithms). Show that the sequence is bounded and strictly increasing.
It is a bit worksome and you need to not to mess up your computations.

HW 5 (due 2/22, 6pm):

Page 36, #8, #10, #14.
Page 43, #5, #6, #8, #11, #12, #13, #15 a, c, #18, #21.

HW 4 (due 2/15, 6pm):

Page 28, #2 a), b), #3, #4, #5, #6 a) and b), #15 (with proof!).
Page 36, # 1 b), #2 b), f), #3, #7.

HW 3 (due 2/7 , 4pm):

Page 21-22, #1, #2 b), #3, # 4 a), #11, #12 c), #13 a) .
# 5, # 7, b), # 8 b), c), #10 b).

HW 2 (due 1/31, 4 pm):

Page 16, #1 a), b), c), f).
#2, #3 d), i); #4, #6, #11 a), d).

Prove that the square root of 3 is irrational. Prove that the square root of 5/7 is irrational.

HW 1 (due 1/24, 4 pm):

Page 6, #2, parts a) and b); #3; #8 parts a) and e); #10 part c).
Pages 11-12, # 3, #5 b),# 6 b), #8 b), c) and f), #9 ( for 8 b), c) and f) ).