Questions tagged [proof-explanation]
This tag is for readers who ask for explanation and clarification of some steps of a particular proof.
10,073
questions
1
vote
0
answers
36
views
Proof from Lang’s Basic Mathematics
Starting on page 11, I don’t understand the proof.
If a, b are negative integers, then a + b is negative.
Proof. We can write a = —n and b = —m, where m, n are positive. Therefore a + b = —n — m = —(n ...
- 11
1
vote
1
answer
32
views
0
votes
0
answers
24
views
Spivak Calculus Ch. 11, Prob. 61a: $f$ diff in interval containing $a$, $f'$ discont. at $a$. Prove one-sided limits of $f'$ at $a$ cannot both exist.
The following is a problem from ch. 11 of Spivak's Calculus
Suppose that $f$ is differentiable in some interval containing $a$, but that $f'$ is discontinuous at $a$. Prove the following:
(a) The ...
- 1,655
1
vote
1
answer
25
views
$f$ cont. at $a$, $f'$ exists in interval containing $a$ (except possibly at $a$), $l=\lim\limits_{x \to a^+} f'(x)$ exists. Does $f'(a)=l$?
This question regards the following theorem (as stated in Spivak's Calculus):
Theorem 7: Suppose $f$ is continuous at $a$, $f'(x)$ exists for all
$x$ in some interval containing $a$, except perhaps ...
- 1,655
0
votes
0
answers
17
views
uniqueness of clutching decomposition Hatcher p.$47$ Lemma $208$
In the following accepted questions on mse (first and second) concerning K theory, in particular the uniqueness of the splitting given also in Hatcher p.$47$ Lemma $208$ is addressed by the same ...
- 5,001
0
votes
0
answers
20
views
Mistake in proof of "double derivative test" in calculus textbook
I'm currently studying for a semester test in advanced calculus, and one of the topics covered is finding the local minima and maxima of a 3 dimensional surface. The first theorem that was proved was ...
- 59
1
vote
0
answers
20
views
Prove that every cycle graph $C_n$ has $n$ edges
I need to prove this directly and by induction. I do not even know where to start.
Question:
A cycle graph $C_n$ is a connected graph with $n$ vertices, such that each vertex is adjacent to exactly ...
- 11
1
vote
1
answer
23
views
Fredholm Index of Toeplitz operators with invertible and continuous symbol
I'm working through the following proof in C* algebras by Murphy, and I'm stuck on a step in the proof. For reference, $\epsilon_n = z^n : T \longrightarrow \mathbb{C}$, and $\Gamma = \text{span}(\...
- 678
0
votes
1
answer
12
views
$x=Px+Qx$ unique decomposition in Hilbert space
Problem:
Let $M$ be a closed subspace of Hilbert space $H$. Then
every $x \in H$ has unique decomposition
$x = Px + Qx$ where $Px \in M$, $Qx \in M^{\perp}$ .
I don't understand: When proving that our ...
1
vote
1
answer
62
views
Spivak's Calculus, Ch 11, problem 59: $f$ has property $(f')^2=f+\frac{1}{f^3}$, find formula for $f''$ in terms of $f$.
The following is a problem from chapter 11, "Significance of the Derivative", from Spivak's Calculus
Redo problem 10-18(c) when
$$(f')^2=f-\frac{1}{f^2}\tag{1}$$
Here is problem 10-18(c)
...
- 1,655
0
votes
1
answer
51
views
Proof of Łoś-Tarski theorem: explanation of the obtained contradiction
I'm going through the following document about model theory: https://webspace.science.uu.nl/~ooste110/syllabi/modelthmoeder.pdf in which a proof of the Łoś-Tarski preservation theorem is given.
I ...
- 83
3
votes
3
answers
48
views
Spivak's Calculus, Ch. 11, "Significance of the Derivative", prob. 58: Prove $f'$ increasing then every tangent line intersects graph of $f$ only once
The following is a problem from ch. 10, "Significance of the Derivative", from Spivak's Calculus
Prove that if $f'$ is increasing, then every tangent line of $f$ intersects the graph of $f$ ...
- 1,655
0
votes
0
answers
30
views
Spivak's Calculus: Understanding proof of alternative form of l'Hôpital's Rule
The following problem is from chapter 11, "Significance of the Derivative", of Spivak's Calculus.
There is another form of l'Hôpital's Rule which requires more than
algebraic manipulations: ...
- 1,655
1
vote
1
answer
59
views
An example showing that a quotient space of an hausdorff space is not hausdorff.
Let be $\mathscr P$ the partition of $[0,1]^2$ defined through the position
$$
\mathscr P:=\big\{\{x\}\times[0,1]:x\in\Bbb Q\big\}\cup\big\{\{(x,y)\}:(x,y)\in \mathbb R \setminus \mathbb Q\times[0,1]\...
- 3,432
2
votes
2
answers
187
views
Understanding a 'geometrical proof' of irrationality of √2
I had been having trouble understanding a proof of the irrational nature of √2.
I found this proof in the first page of the foreward to 17 theorem provers of the world where a 'geometrical proof' (is ...
- 125
1
vote
0
answers
32
views
Doubt from the proof that the sequence space $l^2$ is a Hilbert space from Kreszig book
This is the proof given in Kreszig book.
3.1-6 Hilbert sequence space $l^2$. The space $l^2$ (cf. 2.2-3) is a Hilbert space with inner product defined by
$$
\langle x, y\rangle=\sum_{j=1}^\infty \...
- 572
1
vote
0
answers
15
views
If $\tau ^{(n)}\downarrow \tau $ then $\{X_{\tau ^{(n)}}\}_{n\in \mathbb{N}}$ is uniformly integrable
Reading in the book of Bhattacharya and Waymire of probability theory I find the following assertion:
Let $\{X_t\}_{t\in [0,T]}$ a right-continuous martingale adapted to a filtration $\{\mathcal F_t\}...
- 26k
-2
votes
1
answer
61
views
In the proof of construction of canonical module
I'm reading David Eisenbud's Commutative Algebra, p.539, Theorem 21.15 :
I'm trying to understand the underlined statement. Why is it true?
My first attempt is,
Question 1. Let $I:=\operatorname{...
- 676
3
votes
2
answers
86
views
How to rigorously fill in certain steps in the proof of L'Hôpital's Rule as it appears in Spivak's Calculus?
The following is L'Hôpital's Rule as it appears in Spivak's Calculus
Suppose that $\lim\limits_{x \to a} f(x) = \lim\limits_{x \to a}
g(x)=0$
and suppose also that $\lim\limits_{x \to a} \frac{f'(x)}...
- 1,655
0
votes
0
answers
11
views
Derivation of an upper bound
This is question 24 of this collection of dice problems with solutions, I couldn't understand this
$$\begin{align}
\frac{1+5^n+\max\{1+6,1+C\log(n-1)\}\sum_{j=1}^{n-1}{n \choose n-j}5^j}{6^n-5^n}=\...
1
vote
0
answers
49
views
Prove that if a is an infinite cardinal, then for each finite cardinal n, $a^n = a$
The proof in my textbook is the following:
$a^n = card\:A^n$, the set of all maps of $I_n$ into a set $A$ of cardinal $a$. Since $A^{n} \subset I_n \times A$, $a^n \leq na = a$. On the other hand, for ...
- 61
0
votes
0
answers
20
views
What exactly is being asked in Spivak's Calculus, Ch. 11, "Significance of the Derivative", problem 49, regarding Cauchy Mean Value Theorem?
The following is a problem from chapter 10, "Significance of the Derivative", from Spivak's Calculus:
Prove that the conclusion of the Cauchy Mean Value Theorem can be written in the form
$...
- 1,655
1
vote
0
answers
53
views
How do formal proofs work and relate to interpretations?
As far as I know statements in formal logic are written in a (formal) alphabet which are just symbols, where the allowed sentences have to follow certain rules. If they do, they are called well formed....
- 195
2
votes
0
answers
45
views
Doubt about a geometric solution to Problem $10$ of the $2017$ AIME I contest
The following is Problem $10$ of the $2017$ AIME I contest
Let $z_1=18+83i,~z_2=18+39i,$ and $z_3=78+99i,$ where $i=\sqrt{-1}.$ Let $z$ be the unique complex number with the properties that $\frac{...
- 310
0
votes
0
answers
29
views
Chapter 11, Theorem 5.2 (4) of James Dugundji Topology
Let $p:X \to Y$ be a perfect map. Then: $(4)$ If $X$ is $2^\circ$ countable, so also is $Y$.
Dugundji’s proof:
Let $\{U_i\}$ be a countable basis for $X$, and let $\{V_i\}$ be the family of all ...
- 1,379
0
votes
0
answers
17
views
Proof of small-span theorem in Apostol's Calculus Volume 1
The span of f is defined as follows:
Let $f$ be real-valued and continuous on a closed interval $[a,b]$ and let $M(f)$ and $m(f)$ denote, the maximum and minimum values of $f$ on $[a,b]$. We shall ...
- 1
1
vote
1
answer
26
views
Spivak Calculus: $f$ continuous on $[a,b]$, $n$-times differentiable on $(a,b)$, with $n+1$ roots in $(a,b)$. $f^{(n)}(x)=0$ for some $x$ in $(a,b)$.
The following is a problem from chapter 10, "Significance of the Derivative", in Spivak's Calculus
Suppose that $f$ is continuous on $[a,b]$, that it is $n$-times differentiable on $(a,b)$, ...
- 1,655
0
votes
0
answers
21
views
question on a proof Dirichlet's approximation theorem
We will fix some positive integer $N$ and consider the fractional parts of the numbers $0,α,2α,...,Nα$. Stick them into this collection of $N$ intervals
$$[0;\frac{1}{N}),[\frac{1}{N},\frac{2}{N}),...,...
- 1
2
votes
2
answers
71
views
Spivak Calculus: $f$ satisfies $f''(x)+f'(x)g(x)-f(x)=0$, for some g. Prove that if 𝑓 is 0 at two points, then 𝑓 is 0 on the interval between them.
The following is a problem from chapter 11, "Significance of the Derivative" from Spivak's Calculus
Suppose that $f$ satisfies $$f''(x)+f'(x)g(x)-f(x)=0\tag{1}$$ for some function $g$. ...
- 1,655
0
votes
0
answers
6
views
Showing that the intrinsic mean of continuously distributed data on the unit circle is almost surely unique
The article I am currently reading is Intrinsic Means on the Circle: Uniqueness, Locus and Asymptotics by Hotz and Huckerman, pp. 7. Below is a screenshot of the authors' proof that the intrinsic mean ...
- 259
1
vote
0
answers
22
views
Figuring out a proof that $\mathbb{P}\{-\pi \leq X \leq -\pi + \delta\}=\frac{\delta}{2\pi}+\frac{\delta^{k+1}}{(k+1)!}f^{k}(-\pi+) + o(\delta^{k+1})$
The article I am currently reading is Intrinsic Means on the Circle: Uniqueness, Locus and Asymptotics by Hotz and Huckerman, pp. 4-5.
Suppose that $X$ is a random variable living in the unit circle ...
- 259
4
votes
0
answers
39
views
+200
Husemoller: homotopy of linear clutching map (proposition $4.5$, pag. $187$)
Background :
I'm currently studying vector bundle through the book of [husemoller,"fibre bundles"]
(https://www.maths.ed.ac.uk/~v1ranick/papers/husemoller). The following question concerns a ...
- 5,001
-1
votes
2
answers
71
views
How to prove $\frac{1}{\sqrt{a} + \sqrt{b}} = \sqrt{a} - \sqrt{b}$?
My daughter is learning how to rationalise surds for her school exams. One example being worked through is the following:
$\frac{1}{\sqrt{6} + \sqrt{5}}$
In the tutorials she is following, the first ...
- 229
0
votes
0
answers
44
views
Probability of alternating heads/tails - infinite coin tosses
We toss a fair coin until we get two consecutive heads or two consecutive tails. Each sequence of $n$ coin tosses has a probability of $\frac{1}{2^n}$.
What is the probability that this experiment ...
- 3,061
3
votes
2
answers
107
views
Chapter 11, Theorem 5.2 (2) of James Dugundji Topology
Let $p:X \to Y$ be a perfect map. Then: $(2)$ If $X$ is regular, so also is $Y$.
Dugundji’s proof:
Given $y\in U$ in $Y$, there is by 1.5(b) an open $V \subseteq X$ with $p^{-1}(y) \subseteq V\...
- 1,379
1
vote
1
answer
23
views
Clarification of Bondy's proof that an Ore graph on $n$ vertices has $\geq \frac{n^2}{4}$ edges
This question concerns a proof from Bondy, Pancyclic Graphs 1 of the fact that a graph $G$ on $n$ vertices satisfying the condition that $d(u) + d(v) \geq n$ for any non-adjacent vertices $u,v$ has at ...
- 2,309
1
vote
0
answers
22
views
Existence of conformal equivalence between doubly connected domain and annulus
The following math overflow post
https://mathoverflow.net/questions/261535/mapping-the-doubly-connected-domain-to-an-annulus
provides a sketch proof of the fact that any (non-degenerate) doubly ...
- 67
2
votes
1
answer
51
views
Triple bar meaning in proof of the Principle of Superposition
What does the triple bar mean in this context?
"Thus $y(t)≡0,...$"
- 759
1
vote
1
answer
26
views
Confused on two parts of Proof in C* algebras by Murphy
I'm working through the proof of the following theorem from C* algebra by Murphy. For context, $T_{\varphi}: H^2(T) \longrightarrow H^2(T)$ is given by $T_{\varphi}(f) = p(\varphi f)$ for $p: L^2(T) \...
- 678
2
votes
0
answers
30
views
Proving that an associative binary operation gives rise to a group [duplicate]
I'm trying to prove the following claim.
Let $S$ be a nonempty finite set, equipped with an associative operation $*: S \times S \to S$ such that, for every $x,y \in S$, there exists $z \in S$ such ...
- 231
1
vote
1
answer
51
views
Understanding a proof about $\lambda$-supercompact cardinal
I have trouble understanding the proof of this
Lemma 20.15 from Jech's Set Theory, could someone explain why is $(2^\alpha)^M = (\alpha^+)^M = \alpha^+$?
- 13
0
votes
2
answers
33
views
Artin, Chapter 2, Misc.6
I am trying to solve miscellaneous exercise 6 in Chapter 2 of Artin's book, Algebra. Below is the statement of the problem.
Let $a = (a_1, \ldots, a_k)$ and $b = (b_1, \ldots, b_k)$ be points in $k$-...
- 231
1
vote
0
answers
15
views
Hyperplane arrangement : The Shi arrangement
I have been lately reading Hyperplane arrangement lectures by Richard Stanley on https://www.cis.upenn.edu/~cis610/sp06stanley.pdf . In lecture 5, Theorem 5.16 we define the characteristic polynomial ...
- 11
0
votes
0
answers
32
views
Please help me understand a proof for Taylor's Theorem
When using integration by parts, the proof appears to turn the latter function $(x-t)^k$ into the additive inverse of its integral $-(x-t)^{k+1}/(k+1)$. I don't understand how this could possibly be ...
- 1
0
votes
1
answer
25
views
A sequence towards to the upper bound. [closed]
Let $A\subseteq\mathbb{R}$ be a no empty set with upper bound $m:=\sup{A}\in\mathbb{R}$, then it's simple to prove that exists a sequence $\{x_n\}\subseteq A$ such that $$\large x_n\to m$$ for $n\to \...
- 169
1
vote
1
answer
33
views
Query on Serre's Proof
Hello I was reading Serre's book Linear Representations of finite groups and had a doubt while reading a proof of a proposition (Proposition 24, pg. 61):
Proposition 24. Let A be a normal subgroup of ...
- 1,228
1
vote
0
answers
31
views
Why is $\mathbf{u}=\alpha\ \mathbf{e}_k$?
I've read this answer to a question about Symmetric Rank Update (SR1).
In this approach, we require the update of the Hessian matrix to be of
the form $$ \mathbf{B}_{k+1}=\mathbf{B}_{k} + \mathbf{u}\...
- 1,404
-1
votes
0
answers
22
views
Automata Regular Expression that remembers n iterations
Given is $L = \{\sigma_1 ~u~\sigma_2~v~\sigma_3 ~|~ \sigma_{1,2,3} \in \Sigma,~~ u,v\in \Sigma^*,~ |u|=|v|,~ \sigma_2=\sigma_3 ~or~ \sigma_2=\sigma_3 ~~\mathbb{but ~~ not ~~ both} \}$
I do not ...
- 654
-2
votes
3
answers
111
views
Prove $d(f,g)= \min {|f(x)-g(x)}|$ in $C[a,b]$ is not metric
Trying to answer whether we can set in the set of real continuous functions in the closed interval $[a,b]$ metric $d(f,g)$
I found this
For the min case, let $a=−1$ and $b=1$ and consider the ...
- 301
0
votes
1
answer
21
views
Confused on a set inclusion in C* algebras by Murphy
Im stuck on the following part of this theorem:
The closed vector subspaces of $L^2(T)$ invariant for the bilateral shift $v=M_{z}$ (for $z: T \longrightarrow \mathbb{C}$ the inclusion map) are ...
- 678