How to solve this problem?

Let R stand for the real numbers. Let C stand for the complex numbers. Let K = R or K = C. Let f: K → Kn be a map,n∈N. Let x, h ∈ K. Consider the following statements:

1. Thereisy∈Kn suchthatthemapggivenbyg(0)=y,g(h)=h−1(f(x+h)−f(x)) forh̸=0iscontinuousin0.
2. Thereexistsamapφ:K→Kn andy∈Kn suchthatf(x+h)=f(x)+hy+hφ(h),φ(0)=0,andφiscontinuo us
   in 0.
3. f is continuous at x.

Now, do the following:

1. Discuss why the statement 1 means that f is differentiable, i.e., the tangent-line is the limit of secant-lines!
2. Discuss the connection of statement 2 with Taylor’s formula! You can quote Taylor’s formula from a book/course whichyou know.
3. Show with detailed arguments and the means of our course that statements 1⇔2⇒3. (This shows that differentiablefunctions are continuous).
4. Use statement 2 and estimates for properly chosen converging series to show with the means of our course that theexponential function z → exp(z), z ∈ C is complex (and thus real) differentiable.

: To show that statements 1⇔2 in subtask 3, express φ with g and vice versa. To show that statements 2⇒3, consider


HINT
hν = xν − x where the xν have limit x. To show that the exponential function is differentiable, start with exp(x + h) =exp(x) exp(h) = exp(x)(1 + h + · · ·). Then make a detailed estimate/argument why φ is a converging sum (well-defined) andφ is continuous and φ(0) = 0.

Let xν ∈ Rn, ν, n ∈ N. The sequence (xν )ν∈N is called a Cauchy sequence, if∀ε>0∃νo∈Nsuchthatν1 >νo andν2 >νo imply||xν1 −xν2||<ε
i.e., elements in the tail of the sequence are (arbitrarily) close together. Now, do the following:
1. Let n=1, i.e., xν ∈ R. Show with the means of our course that every Cauchy sequence in R as above has a limit
x ∈ R. That means: for every Cauchy sequence (xν)ν∈N in R, there exists x ∈ R such that x = limν→∞xν.
The above means for example that π exists, since π’s finite-digits rational approximations 3, 3.1, 3.14, 3.141, 3.1415.. .
form a Cauchy sequence (provided it can be determined).
2. Use subtask 1 above to show with the means of our course that every Cauchy sequence in Rn as above has a limitx ∈ Rn. That means: for every Cauchy sequence (xν)ν∈N in Rn, there exists x ∈ Rn such that x = limν→∞xν. Inmathematical terms, you have shown that Rn is a Banach space.



Attachment 48833Attachment 48833: To prove subtask 1, you can employ our theorem about the existence of limsupν →∞ xν and liminf ν →∞ xν . For the proof


HINT
of subtask 2, apply a result of our course about component-wise convergence.

Let x = (x1,. xn) ∈ Rn, n ∈ N, and defineand
nν=1

x2ν)1/2||x||∞ = max (|xν |).
ν =1,. n


||x||2 =(


Define
1. Show with detailed comment and the means of our course using more or less direct proof from the definition, or quota-


Bk(x,r)={y∈Rn:||y−x||k <r}, k∈{1,2,∞}.tions of results from our course that both || · ||2 and || · ||∞ are norms on Rn.


(n)
2. Find constants cν,μ such that

||x|| ≤ c(n)||x||ν ν,μ μ


for every x ∈ Rn, ν, μ ∈ {1, 2, ∞}. (6 formulas). Provide detailed comment and more or less direct proof from thedefinition.

1. Use subtask 2 and the means of our course to show that Bk (x, r) is open for k ∈ {1, 2, ∞}.
2. Use subtask 3 to show with the means of our course that (0, 1)n is open in Rn.

: For the proof of subtask 1, you can use that the absolute value in R is a norm which coincides with ||x||2 (why?). For


HINT
the proof of subtask 2, draw pictures detailing the situation for n = 1, 2, 3 as a helpful guide.

Attachment 48834

Show with detailed comment and the means of our course that
1. U = (0, 2017) × (0, 2017) × (0, 2017) is an open subset of R3.


2. A = {(x1,x2,x3,. x17) ∈ R17:(x20+x20+x20+x20+x20+x20+x20+x20)·(x17+x17+x17 +x17+x17+x17+x17+x17+x17) = 2017}


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17is a closed subset of R17.


3. K = {(x1, x2017) ∈ R2017: 1008 |xk|1/20 = 2017 |xk|17 ≤ 2017}k=1 k=1009
is a compact subset of R2017.
4. Letx,xν ∈Rn,ν,n∈Nsuchthatx=limν→∞xν.LetL={xν:ν∈N}∪{x}.Show thatLiscompact.


: For the proof of subtask 1, you can (a) employ a suitable (continuous?) linear map, (b) employ projections p1,


HINT
p2 and p3 onto the components of R3, or (c) consider the relationship between open sets and converging sequences.For the proof of subtask 2, you can (a) write A as the inverse image of a certain set under a continuous function, or (b)consider the relationship between closed sets and converging sequences. For the proof of statements 3 and 4, you mayuse a characterization (the only!) of compact sets in Rn from our class.





Attachment 48835

Let p: [0,1] → [0,1] be given by
p(x)=(x2 +x3 +x5 +x7 +x11 +x13 +x17)/7=floor(x)+p(x−floor(x))
.Letf:R+ →R+ begivenby
f (x) = floor(x) + p(x − floor(x)).Examples: floor(0) = 0, floor(0.4) = 0, floor(4) = 4, floor(4.4) = 4.
1. Show with detailed comment the means of our course that f is a well-defined, continuous function.2. Show with detailed comment the means of our course that f is bijective.
3. Show with detailed comment the means of our course that f −1 is a continuous function.: For the proof of subtask 1, you can prove by induction that f is continuous on [1 − n, n] for n ∈ N. The Induction-


HINT
Foundation n=1 may require some work. In the Induction-Step n ⇒ n + 1, you can use the Glueing Lemma from our course.For the proof of subtask 2, you show and use that f is strictly monotone increasing. And you can use the Intermediate ValueTheorem. For the proof of subtask 3, consider a converging sequence in R and restrict f to an interval [−N,N] where N islarge enough (for what?). Then, observe that [−N, N ] is compact. This argument is similar to the proof in our course that thesquare-root (and other root-functions) is continuous.









Attachment 48836

Let C stand for the complex numbers. Let S1 ⊂ C be the unit-circle in the complex plane. Let z ∈ S1. For n∈N, letpn(z) = zn. Consider the polynomial p(z) = (z − 3/2)2(z − 1/2)2z(z + 1/2)2(z + 3/2)2. Set f(z) = p(z)/|p(z)|.

1. Show with detailed comment and the means of our course that f: S1 → S1 is a well-defined, continuous function.
   
2. For some n∈N, find a continuous function F: S1 × [0,1] → S1 such that F(z,0) = pn(z), and F(z,1) = f(z). Show
   with detailed comment and the means of our course that F is continuous and that F satisfies the latter identities.
   
3. Is the n found in part 2 uniquely defined? Give a detailed reason for your answer! (with the means of our course
   
4. Develop a theory with the means of our course which says that the homotopy-class of f as above for a polynomial p
   with no zeroes in S1 counts the zeroes of p within S1. Use the Fundamental Theorem of Algebra as a tool in your proofso that you can define a homotopy similar to the above.

: For the proof of subtasks 3 and 4 use the main Theorem about H1[S1] in our class stating H1[S1] ≡ Z, the


HINT
integers, and [pn] ≡ n.





Attachment 48837

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