## Homework assignments, Real Analysis I - Math 446, Fall 2014

Dates listed are assignment dates, not due dates; homework should be done by the next class meeting after the assignment date.

#### Prof. Janeba, Fall 2014

Homework assignments for the week of:
Aug. 27 | Sept. 3 | Sept. 8 | Sept. 15 | Sept 22 | Sept 29 | Oct 6 | Oct 13 | Oct 20 | Oct 27 | Nov 3 | Nov 10 | Nov 17 | Nov 24 | Dec 1
Final Exam
Interesting stuff relevant to our class

### Week #1:

• Aug. 26:
• Read syllabus (quiz next class)
• Study chapter 1.1 and 1.4 of our text (possible open-notes reading quiz next class)
• Do chapter 1.1 #2ac, 3b, 5a  and chapter 1.4 #3ab, 5abc, 8, 12
• Aug. 28:
• Do 1.4 #1a (hint: find some way to use the distributive law), 13,14,15
• Do 1.2 #3,4,11
• Suppose that f is a function from A to D, and B and C are subsets of A.  Prove that if B C then f(B)f(C).
• Suppose that f is a function from A to D, and E and G are subsets of D.  Prove that if E G then f -1(E)f -1(G).
• Prove that if a,b with a<b then a < (a+b)/2 < b.
• Note: If A={1,2,3,4} then sup A = 4.  If B = [2,4] then sup B = 4.  But if U = (0,), then no sup for U!

### Week #2:

• Sept. 2:
• Catch up
• Show, using only the existence of the set P of positives from p. 20-21, and its properties (O1) and (O2), that if a<b and b<c, then a<c, the transitivity property of inequalities.  [I'm surprised this isn't mentioned in Stoll!]
• Expand on the proof of theorem 1.2.5(b) in the text (the text version is very concise).
• Do 1.2 #8,9
• Sept. 4:
• Show that if f:AB and EB, then f -1(EC) = [f -1(E)]C.
• Finish up the in-class exercise: Write up a proof that f:AB is a one-to-one function if and only if
for all A1, A2A f (A1 A2) = f (A1)f (A2) .  This write up IS done in groups.
• Read 1.5 & 1.6, and start on chapter 1.5 #1,3,4,5,6a
• Turn in next class meeting:  chapter 1.1 #5a, 1.2 #9,11, 1.4 #5abc,12, and the write-up for the in-class exercise above.

### Week #3:

• Sept. 9:
• Do 1.5 #7; you may use the lemma that 2n > n n without proving the lemma.
• Give decimal values for each of: 0.1010(2),  0.101010...(2), which repeats the block "10" forever [I don't have a way to "overline" in html], and
0.1010(3),  0.101010...(3), which also repeats the block "10" forever.
• Find the ternary (base 3) expansion of 1/4 (recall that 4(10) = 11(3) ).  You can do this by long-division in ternary
or by "building up" 1/4 from negative powers of 3, e.g. "How many 1/3's fit in 1/4, then how many more 1/9's, then how many more 1/27's" and so on.  Show work somehow.
Hint:  the expansion will repeat.
• Check your answer for 1/4 above by working out a geometric series calculation.
• Now find a ternary expansion of 1/7 (recall that 7(10) = 21(3) ).
• Sept. 11:
• Do 1.7 #1,3 (hint for 3 - every natural number has a unique factorization into primes.  That should get you started.)
• Do 1.7 #17 (hint:  Find a first point in (-,) - then note why the point you found can't be the largest such point.  Then find another, and continue.)
• Give a direct proof that the union of two disjoint, countable sets is countable.  It may be convenient to use enumerations here.
• Use the prior problem to do 1.7 #13
• do 1.7#14 by using 1.7.7

### Week #4:

• Sept. 16:
• Show that if A~B  and C~D, then A×C ~ B×(recall A~B means there is a bijection between these two sets).
• Show that ×(×) is countable, finding a way to use the prior problem and another result established in class (don't reinvent the wheel here!)
• The set of zero-degree polynomials with integer coefficients is countable because: (give your answer, then:)
• Every 1st degree polynomial with integer coefficients can be formed by adding mx (for some m in the integers) to a zero-degree polynomial.  Explain why, then use this to prove that the 1st degree polynomials are countable.
• Show that for any n in , the nth degree polynomials with integer coefficients are countable
• Show that the set of all polynomials with integer coefficients is countable (note that polynomials necessarily have finitely many terms).
• Make sure you know how to do 1.4 #15 (that inf/sup problem I mentioned in class).
• Sept. 18:
• Quiz 1
• Prove that whenever a real number in [0,1] has two ternary expansions, at least one of them contains the digit "1", so if a number is in the Cantor set, it has only one expansion consisting of only 0's and 2's.  (This should be short).  You may take as given the fact that if a number has two ternary expansions, one of them ends in a string of repeating 2's.
• Bonus:  Find a number in the Cantor set that is not an endpoint of one of the removed subintervals.
• Read chapter 2.1 in Intro. to Real Analysis (our regular textbook).
• Do 2.1 #2,3,6a,b, start to think about #8a.
• Read p.1-2 of the Notes on Metric Spaces handout from class (also available on the class WISE site).
• Do problems 1-3 on p.2 of the handout.

### Week #5:

• Sept. 23:
• Read 2.1 & 2.2, do 2.1 #8abcd, 9ab, and 11a (use the hint in the answer key OR the fact that if x,y are positive, x<y iff log x < log y), start to think about 15.
• Start also to think about 2.2 #1
• Sept. 25:
• Read 2.3, do 2.1 #15 and 2.2 #1, 2,3, 6acd (use limit theorems on #6, rather than making an -N  proof), 7abcd,10

### Week #6:

• Sept. 30:
• Prove that if ann  and anL, then LM.
• Do 2.2 #6befgh
• Spend some serious time summarizing/outlining the material we have covered this semester, in particular, identifying concepts, definitions, theorems, and above all, problems that we have studied.
• Spend about twice as much time actually doing problems and testing yourself for recollection of theorems and definitions.
• On Thursday, turn in: 2.1 #6b,8b,15, 2.2#2a,6a, 6e and the "LM" problem above in today's assignment.
• Be sure, for each problem, to use only those methods we had established at the time the problem was assigned.  That is, you may not use chapter 2.2 theorems to solve a chapter 2.1 problem.
• Oct. 2:
• Exam 1

### Week #7:

• Oct. 7:
• Do 2.3 #3abc,4,5,7a,c,9,12a,b, and try 17ab
• Do 2.4 #3a,b,d,e
• Oct. 9:
• Read section 3 of the Metric Spaces handout
• In Exercise set 2 of the metric spaces handout, do #1,2,5,7 (#5 should help for #7)
• Prove that given a sequence {an} with positive terms, an  iff  1/an0.
• Prove that if ana, bnb, and anbn  n, then ab
Hint:  Apply previous work to {bn - an}
• Do 2.4 #7abde, and think about how hard #12 is.

### Week #8:

• Oct. 14:
• To prove that every sequence in has a monotone subsequence, we started with case 1: Every tail end has a largest (maximum) element, and proved that such a sequence contains a decreasing subsequence.
• Prove the "other" case also leads to a monotone subsequence.
• Use the "lemma" above and another major theorem we proved in class to prove the Bolzano-Weierstrass theorem:  Every bounded sequencein has a convergent subsequence.  This proof should be short!
• Do 2.4 #5,8,9 (9 seems hard; see what you can do with it)
• Oct. 16:
• Do 2.5 #1abce (Give some reasoning, even if not a full proof of your answers)
• Do 2.5 #2,4,5a,8
• Oct. 17:  Mid-semester day - no classes

### Week #9:

• Oct. 21: Do chapter 2.6 #3, and catch up
• Oct. 23:  Quiz

• Oct. 28:
• Oct. 30:

• Nov. 4:
• Nov. 6:

### Week #12:

• Nov. 10: Take-home exam (exam #2) due at 3 p.m.
• Nov. 11:  Read 3.2, and sections 6 and 7 of the Metric Space Handout (on WISE); read especially carefully theorem 19 & its proof on p.12-14, do 3.2 #1,3,4a,8,10b  (see p. 69 for def. of isolated point)
• Nov. 13:  Prove the following:
• If xnxo in a metric space (X,d), then the only (possible) limit point of {xn : n} is xo(and maybe not even that; just prove nothing else could be a limit point.)
• (This is really exercise #3 on p. 15 of the metric space handout; do either this or #3)
• Prove that totally bounded sets are bounded. (like, totally, dude).
• #5,8 on metric space handout, p.15
• Note that with #8 above, and prior work, we will have proved that a set K is compact if and only if every sequence in K has a subsequence converging to a point in K.

### Week #13:

• Nov. 18:
• Given a set E in a metric space (X,d), every infinite subset S of E has a limit point in E iff every sequence in E has a subsequence converging to a point in E.
• Note 1: part of this we have already done in #8 last time.
• Does it make a difference if E is finite?   (Note the problem above is worded correctly).  Make sure your proof(s) reflect this consideration.
• Give an - proof that if the limit of f(x) as xa is A, and the limit of g(x) as xa is B, then the limit of f(x)+g(x) as xa is A+B.
• That's different than the proof in the text!
• Do 4.1 #1ab,2ace,6,7a,9,10
• Nov. 20:

### Week #14:

• Nov. 25:  Read 4.2 some more, and do 4.2 #1b, 5,9a,11a
• Turn in next Tuesday:
• 4.1 #2e,9
• Give an - proof that if the limit of f(x) as xa is A, and the limit of g(x) as xa is B, then the limit of f(x)+g(x) as xa is A+B.  Note an - proof means "not using sequences."
• Prove that a compact set K in has a maximum element, i.e. mK s.t. xKxm.  Hint:  an easy proof involves words including bounded, sup, limit point.
• Nov. 27:  Thanksgiving Break, no classes

### Week #15:

• Dec. 2: 1st two problems on the take-home final (due 5 p.m. Thurs. 12/11):
1. Given a metric space (X,d), a compact set KX, and a point xoX \ K, prove that there is a point y in K nearest to xo, i.e. that d(xo,y)d(xo,x) for all xK.
Hint:  Let f : X be given by f(x) = d(xo,x); prove f is continuous and go from there.
2. If f is increasing on [a,b] and maps [a,b] onto [c,d] (where a,b,c,d are real numbers) then f is continuous on [a,b].
Note:  While the statement is true as given, you should only prove continuity at points x such that f(x)(c,d).  The remaining cases are just variations on the same theme, and I won't ask you to do them.
There are plenty of fussy details still here, some so (seemingly) obvious that you might forget to prove them.  Don't.
3. More problems to come this week.
• Dec. 3: Here are some more take-home problems:
1. Recall the theorem, proved in class:
2. Theorem:  If (X,d) and (Y,ρ) are metric spaces and f:E→Y, E⊂X, and p is a limit point of E, then
lim (as xp) f(x)=L  iff  ∀ sequences xn in E s.t. xnp, f(xn)→L.
a. Use the theorem to prove this corollary:
Corollary:   If (X,d) and (Y,ρ) are metric spaces and f:E→Y, E⊂X, and p is a limit point of E, then f is continuous at p iff ∀ sequences xn in E s.t. xnp, f(xn)→ f(p).
b.Use the theorem and/or corollary above to prove:
Corollary:  If X,Y,Z are metric spaces with A⊂X, B⊂Y and f:A→Y, g:B→Z [and f(A) ⊂B, so that g∘f is defined on A], and f,g are continuous then g∘f is continuous.
3. Re-prove the corollary in 3b using the "continuous inverse images of open sets" characterization of continuity.
4. Show that if f is continuous from (X,d) into , that {xX : f(x)=0} is a closed set.
• Dec. 4:   More take-home problems
1. Prove that if f is continuous on (a,b] with a≠b and lim (as xa) f(x) exists, then f is bounded on (a,b].  Hint:  If the interval had been closed, this would be much easier - can you get any help from that hint?
2. Given a metric space (X,d), and points x,y,zX, prove that |d(x,y)-d(x,z)|d(y,z).  Hint:  We did something a lot like this with distance in .
3. Suppose that y is a limit point of set E in a metric space (X,d).  Give a careful, step-by-step construction of a sequence in E converging to y (i.e. prove that such a sequence exists by showing how to construct it; prove its convergence also).  Be careful with the fussy details here.
4. A review problem:  Prove that if f:A→B and U,V are subsets of B, then
f -1( U ∪ V ) = f -1( U ) f -1( V ).
5. Give an ε-N proof that {n/(2n²-1)} converges to 0.
6. Suppose that (a,b) with p(a,b), and f:(a,b) is continuous at p.  Prove that if f(p)>0, then there is a neighborhood of p on which f is positive.  Hint:  We just did something awfully close to this.
7. Suppose f:[a,b]→ is continuous, f(a)>0, and f(b)<0.  It is an easy application of the Intermediate Value Theorem that f has a root in (a,b), i.e. that ∃c∈(a,b) s.t. f(c)=0.  Now in class, we proved the I.V.T. using a fairly abstract approach with connected sets and metric spaces.  For this problem, we will give a proof more in line with 19th century mathematics.
a.Let A={x∈[a,b] : f(x)>0}.  Explain why A is nonempty and bounded above.
b.We know what we expect the graph of f to look like.  That expectation makes the desired result "obvious," but of course we want to be more rigorous than that.  However, take a moment to sketch a simple example of a possible graph of f and note where the set A is in your sketch.  Now guess/define a value c (in terms of a supremum), inspired by your sketch, that will probably have the desired properties.
c.Prove that f(c)=0 and that c∈(a,b).  Use nothing beyond elementary properties of suprema, properties of continuity, and problems on this exam.  Hint:  Suppose f(c) weren't zero.  There are two possibilities.
8. That is all; there will be no more problems added.

### Week #16:

• Thursday, Dec. 11, 2014 5 p.m.   Final Exam Due (turning it in early is certainly allowed).

#### Interesting stuff relevant to our class:

More to come, please check back.