### Homework assignments, Real Analysis II (Math 447)

#### Prof. Janeba, Spring 2015

Homework assignments for the week of:
Jan. 19 | Jan. 26 | Feb. 2 | Feb. 9 | Feb. 16 | Feb. 23 | Mar. 2 | Mar. 9 | Mar. 16 | Mar. 30 | Apr. 6 | Apr. 13 | Apr. 20 | Apr. 27

### Week #1:

• Jan. 20: The five-old-familiar-proofs handout - turn in all five on Thursday, with at least one of the five in LaTeX.
• Jan. 22:
• Read chapter 4.2, 4.3.  Do 4.2 #11a, 4.3 #2b,3a,11a,5 (recommended in that order)
• Optional lemma that may prove useful:  If f is continuous on [a,b) and f is uniformly continuous on (a,b) then in fact f is uniformly continuous on [a,b).

### Week #2:

• Jan. 27:
• Do 4.3 #4a, 6a, 10;  10 might be tricky, but think about the graph of f in the way we did with f(x)=(1-x2).
• Read 4.4, do 4.4 #2a, d (here "[y]" means the greatest integer less-than-or-equal-to y, i.e. round down to the nearest integer), 10
• Jan. 29:
• Review problem: Give an - proof that [f(x)+g(x)] = f(x)  + g(x), provided the latter limits exist.  Hint:  Start by assuming the "provided" part, and give 'em names.  For simplicity, you may assume f & g are defined on .
• Using the text notation on p. 148 for one-sided limits, prove the assertion there that:
• Assuming f is defined on an interval I with interior point a, f(x) exists iff  [ f(a+) and f(a-) both exist AND are equal  ].
• Do 4.4 #13 (easy), 15 (hard: use a really big theorem from Real I)
• Do p. 162 #1 (this will take some work.  Draw a simple example first and draw where the cn's are for the first few values of n.  It's fairly easy to show {cn} converges by Cauchiness.
Then show |f(c)-0|< for arbitrary , using |f(c)-0||f(c)-f(cn)|+|f(cn)-0| and a carefully-chosen cn.)  P.S. - this is called the midpoint method of root finding; it is easy to code a computer program to find roots using this algorithm.

### Week #3:

• Feb. 3:
• Give a formal, - proof of the lemma:   Suppose F : E with E  and E containing some neighborhood of a.  Then F(x) = if and only if limh->0 F(a+h) = L.  [The last limit is the limit as h goes to 0 of F(a+h); this is something like our F&G lemma from class today; little computation is needed, but be very precise with the quantifiers.]
• Given xn = 1+(-1)n/and cn=1/2n n, and defining f on [-1,2] by f(x) = cnI(x-xn), sketch the graph of f on [-1,2] at least for points not very close to 1.  Make sure to include at least a few precise pieces or segments of the graph, and try to convey the overall shape.  ALSO, using the theorem from class/text that discussed this construction, tell where f is continuous (no need to prove this beyond correctly citing the theorem).
• Read 5.1, do 5.1 #1abc,2 [for #2, you can use the text hint, or use theorem 5.1.5 and induction]
• State a correct variation of today's F&G lemma for one-sided limits.  No proof needed if the lemma is correct.
• Do 5.1 #14.  At some point, it may be useful to use the lemma that lim h->0 F(h) = lim h->0 F(-h)  [it matters here that h is going to zero!]
• On Thursday, turn in: 4.3 #3a, 11a, the first problem listed for Feb. 3, and 4.4 #15
• Feb. 5:
• Give a formal proof of the lemma: Suppose G:(a,b)→ with (a,b) some interval in , and for some δ>0 and some constant M, G(x)≥M  ∀x∈(a,a+δ).
a.Prove that limxa+ G(x)≥M, if this limit exists.
b.State a mirror-image result involving a left-sided limit (no proof required).
• Do 5.1 #5a, 9 (theorem 4.1.9, the squeeze theorem, may prove useful).
• Examine this list of presentation problems and send me preferences, if any, for problems and partners.  Groups will be assigned soon via email, presentations to start on Feb. 12.
• Catch up!

### Week #4:

• Feb. 10: [Prepare your assigned presentations]
• Feb. 12: Presentations from this list

### Week #5:

• Feb. 17:  do 5.2 #3a,b, 6,13,15  [Hint for 6:  Think about |f '(c)| for c∈(a,b), using the definition of f '.  ]
• Feb. 19:
• Prove the lemma:  limx→0+ f(1/x) = L iff limxf(x) =
(assuming f defined on (0,), and the latter limit defined by 4.1.11 or as defined in class; use either definition as you prefer, they are equivalent by setting  M=1/.)
• Do 5.2 #20
• Read 6.1 through 6.1.5 and skim 6.1.6.
• Do 6.1 #1a
• For next Tuesday, turn in:
• The second bullet item from Feb. 5, i.e. limxa+ G(x)≥M
• 5.1 #5a  (be rigorous, one way to do this uses the item above.)
• 5.2 #6,13 (use the Mean Value theorem to do #13)

### Week #6:

• Feb. 24: Exam I
• Feb. 26:
• Read the rest of 6.1,
• do 6.1 #2a (see example 6.1.6b), 3a, 6,7,13

### Week #7:

• Mar. 3:
• Here's the useful lemma I mentioned in class today; we'll prove a simplified version.  Suppose f is integrable on [a,b] and g(x)=f(x) for x∈(a,b], but g(a) = f(a)+k for some real number k.
Show that g is also integrable on [a,b] and that f = g. You may restrict your proof to the case that k>0, just to save work.
Hints:  Riemann's integrability criterion (6.1.7) will be invaluable in both directions here.  Since f is integrable, what does it tell us (for starters)?
Consider M1 and m1 for f  and M'1 and m'1 for g, and compare them (there are two cases).  Also consider refining the partition by adding a point near a.
• Do 6.1 #4c, 17,21
• Mar. 5:
• Do 6.2 #1,3a,b,4,10.  Note:  Use the theorems of this chapter whenever possible, [not for #1, it is a theorem in the chapter] and the lemma above (Mar. 3)
• Do 6.3 #1.  You don't get to use 6.3.4 here (why not?), but looking at its proof will help; you may assume f is bounded.

### Week #8:

• Mar. 10
• Do 6.3 #2ab,4,5(esp. 5d), 8,11ac, 17ab
• Mar. 12

### Week #9:

• Mar. 17
• Mar. 19

Spring Break Mar. 23 - 27

### Week #10:

• Mar. 31: Prepare group presentations from this handout for Thursday.
• Apr. 2: Read chapter 7.1, do 7.1 #1,2acdfghkn, 4abc, 10

### Week #11:

• Apr. 7:
• Do 7.1 #9,12,11 (suggested in that order, note the terms are not rational functions of k), AND 14 (oops, I misspoke, Becca), and
• Apr. 9
• Do 7.2
• #1,2 (you will need to add quite a bit to the hints in the text),
• #3 (careful, this one is tricky.  Hint:  It makes a big difference whether the terms are non-negative or not.)
• #5a,c,d,e,f

### Week #12:

• Apr. 14
• Apr. 16
• Do 7.3 #2(you'll need to work with partial sums here),3,5,6a,b,c,f,g,9,10,11
• Read 8.1, do 8.1 #1a,b,c (try to be rigorous here), 2

### Week #13:

• Apr. 21
• Do 8.1 #5 (and think what this example illustrates about the different types of convergence we have studied so far)
• Do 8.2 #1 (a variant of something proved in class), #2ab, and think about 2c.
• Do 8.2 #5, 8abc (8.2.7 will work on all of these, though sometimes that will take some real work)
• Do 8.2 #12
• Apr. 23: Quiz at the end of the period

### Week #14:

• Apr. 28
• Read 8.4, skim 8.5, noting the theorem we proved today.
• Do 8.3 #4,  8.4 #5 (note typo in text, the lower limit of all the integrals should be "a", not "0".) You will need important inequalities we have proved for integrals.
• Suppose f is continuous on [0,1] and 0<b<1, show that {f(xn)} converges uniformly to the constant function f(0) on [0,b].
• Do 8.4 #6; the previous problem will help, if we break the integral on [0,1] into integrals on [0,b] and [b,1], and choose b carefully
• Do 8.4 #9, 8.5 #1,3
• [At least a few of these are likely to turn into takehome problems]
• Apr. 30

### Week #15:

• May 5 (Last day of class)
• Friday May 8, 2-5 p.m..  Final Exam