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

- 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:
- Read chapter 1.2, 1.5
- 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!

- 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}(E^{C}) = [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 A_{1}, A_{2}A, f (A_{1}A_{2}) = f (A_{1})f (A_{2}) . 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.

- Sept. 9:

- Do 1.5 #7; you may use the lemma that 2
^{n}> 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) }). - Read chapter 1.6

- Sept. 11:

- Read 1.7, 2.1
- 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

- Sept. 16:

- Show that if A~B and C~D, then A×C ~ B×D (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.

- 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

- Sept. 30:

- Prove that if a
_{n}M n and a_{n}L, 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

- Oct. 7:

- Read 2.4 (& re-read 2.3 about nested intervals) Read section 2 of the Metric Spaces handout (also on WISE)

- 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 {a
_{n}} with positive terms, a_{n}iff 1/a_{n}0. - Prove that if a
_{n}a, b_{n}b, and a_{n}b_{n}n, then ab.

Hint: Apply previous work to {b_{n}- a_{n}} - Do 2.4 #7abde, and think about how hard #12 is.

- 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)

- Read 2.5

- 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
- Read 2.6

- Oct. 17: Mid-semester day - no classes

- Oct. 21: Do chapter 2.6 #3, and catch up

- Oct. 23: Quiz

- Oct. 28:

- Oct. 30:

- Nov. 4:

- Nov. 6:

- 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 x
_{n}x_{o}in a metric space (X,d), then the only (possible) limit point of {x_{n }: n} is x_{o}. (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.
- Read text, chapter 4.1

- 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:

- Do 4.1 #12,14,17a,c (read about limits at infinity in 4.1)
- Read 4.2

- 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. xK, xm. Hint: an easy proof involves words including bounded, sup, limit point.

- Nov. 27:
**Thanksgiving Break, no classes**

- Dec. 2: 1st two problems on the take-home final (due 5 p.m. Thurs. 12/11):
- Given a metric space (X,d), a compact set KX, and a point x
_{o}X \ K, prove that there is a point y in K nearest to x_{o}, i.e. that d(x_{o},y)d(x_{o},x) for all xK.

Hint: Let f : X be given by f(x) = d(x_{o},x); prove f is continuous and go from there.

- 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. - More problems to come this week.
- Dec. 3: Here are some more take-home problems:
- Recall the theorem, proved in class: 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
- Re-prove the corollary in 3b using the "continuous inverse images of open sets" characterization of continuity.
- Show that if f is continuous from (X,d) into , that {x∈X : f(x)=0} is a closed set.

- Dec. 4: More take-home problems
- Prove that if f is continuous on (a,b] with a≠b and lim (as x→a) 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?
- Given a metric space (X,d), and points x,y,z∈X, prove that |d(x,y)-d(x,z)|d(y,z). Hint: We did something a lot like this with distance in .
- 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.
- 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 ). - Give an ε-N proof that {n/(2n²-1)} converges to 0.
- 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.
- 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. - That is all; there will be no more problems added.

lim (as x→p) f(x)=L iff ∀ sequences x

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 x

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.

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

Prof. Janeba's Home Page |

Department of Mathematics | Willamette University Home Page