This tag is for readers who ask for explanation and clarification of some steps of a particular proof.

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29 views

Proof by induction of Gronwall's inequality

I've an exercise which is the following: Gronwall’s Inequality Let $A > 0, B \geq 0$. Let $(\epsilon_j)_{j \in \mathbb{N}}$ be a sequence of real numbers with $$|\epsilon_{j+1}| ...
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0answers
33 views

Goldbach's conjecture. [on hold]

Can someone please say which approaches were used to try to prove Goldbach's conjecture? And please send a link if possible. Thank you in advance.
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1answer
14 views

Proof of discrete probability monotone convergence

I am trying to show that for a sequence of random variables defined on a sample space $\Omega$ $$0\leq X_1(\omega)\leq X_2(\omega \leq ......\leq X_{n}(\omega)...$$ for all $\omega\in\Omega$, with ...
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2answers
18 views

Using the definition of the convergence of the sequence, how can I prove that this sequence converges to the limit?

I suppose what I am confused most on here is the algebraic method. $\lim_{x\to\infty} $ $ 2n^2/(n^3 + 3) $ = 0 I have set it up so far: Let $ \epsilon > 0 $ be given. I set up the equation ...
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2answers
21 views

Understanding a certain step in a proof about a basis of a vector space

This is a theorem from Roman's textbook "Advanced Linear Algebra"(p.$48$). Theorem $1.9.$ Let $V$ be a nonzero vector space. Let $I$ be a linearly independent subset of $V$ and let $S$ be a ...
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2answers
22 views

Construction of a finite projective plane of order $p$, for any prime $p$

I have this construction of a finite projective plane (FPP) of prime order $p$, but I am not sure what's going on. We have already proved that FPPs of order $q$ have $q^2+q+1$ lines and points (if ...
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0answers
32 views

Understanding a spectral theorem's proof

I want to understand the proof of the following theorem: "If $f$ is a self-adjoint operator of a euclidean space, then $f$ is diagonalisable" It goes like this : There are 2 steps that derive ...
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4answers
74 views

A proof that $0=1$? [duplicate]

I saw a proof in a comment on a previous MSE question yesterday and I can't stop thinking about it. It looked like the following: $0 = (1-1) + (1-1) + \cdots = 1 + (-1+1) + (-1 + 1) + \cdots = 1 + 0 ...
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2answers
28 views

if f is a function with domain $\mathbb{R}$ that can be written $f = E + O$ where E is an even function…

if f is a function with domain $\mathbb{R}$ that can be written $f = E + O$ where E is an even function and O an odd function, prove that writing $f$ in this way is unique. Sol'n: A solution that I ...
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0answers
40 views

Question about proving Cauchy's integral theorem

In my course we were given several proofs of Cauchy's theorem, each at various points in the course, each version stronger than the previous. I'd like to learn a proof of the theorem, so naturally ...
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2answers
39 views

How to prove facts regarding sentential logic

Recently I have been very fascinated by the claim and impact of Godel's incompleteness theorem. To understand the proof given by Godel, I felt the need to read an introductory book in logic to begin ...
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1answer
24 views

Why is the tangent space to the orbit through $p\in\mu^{-1}(0)$ an isotropic subspace of $T_pM$?

I'm reading symplectic geometry notes by Ana Cannas da Silva. The set up is a Hamiltonian action $G\curvearrowright(M,\omega)$ of a Lie group $G$ on the symplectic manifold $(M,\omega)$, with moment ...
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2answers
47 views

Understanding a proof of a theorem from S.Roman's “Advanced Linear Algebra”

There is a Theorem $1.5$ on page $43$ of the book "Advanced Linear Algebra" by Steven Roman. Theorem $1.5$. Let $F = \{ S_i | i \in I \}$ be a family of distinc subspaces of a vector space $V$. ...
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2answers
78 views

Proof in Spivak's Calculus

I'm working through Spivak's Calculus, and am trying to show that the following is true: $$ f(x) = \frac{1}{1 + x} \to f( c \cdot x ) = f(x) \text{ for all } c \in \mathbb{R} $$ The book just says: ...
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0answers
20 views

Principal value integral of complex exponential

I'm reading the article Brownian distance covariance and stumbled upon a equality I can't seem to derive myself. We are first presented with the following lemma: and after stating this lemma, the ...
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1answer
59 views

which odd integers $n$ divides $3^{n}+1$?

I don't understand this solution to this problem. Can anyone explain why d divides n?
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1answer
20 views

Eigenspace proof [closed]

I would like to know how to prove that the set of all v satisfying the equation $T(v)=\lambda v$ is a subspace of V. I assume I need to show the subspace axioms: v is non-empty. v is closed under ...
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0answers
10 views

Give an example showing Ran(R1 ∩ R2) ⊆ Ran(R1) ∩ Ran(R2) may not hold as an equality.

I have managed to prove Ran(R1 ∩ R2) ⊆ Ran(R1) ∩ Ran(R2), but I am having trouble finding an example that shows it doesn't hold as an equality.
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3answers
27 views

Let $G$ be a graph with $n$ vertices where every vertex has a degree of at least $\frac{n}{2}$. Prove that G is connected.

First question, if the problem uses a fraction such as $\frac{n}{2}$, would we round down in case $n$ is odd? As for the actual problem, I'm trying to do this with induction and contrapositive and ...
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0answers
75 views

Prove that there exists at least one root of $g$ between any two roots of $f$ [duplicate]

Given that $$f(x)= 1 - e^x\sin(x)$$ $$g(x)= 1 + e^x\cos(x)$$ Using Rolle's theorem, prove that there exists at least one root of $g$ between any two roots of $f$. Attempt so far: $f'(x) = ...
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0answers
13 views

Continuous time fourier transform existance proof explanation

The continuous time fourier transform,$$X(jw) = \int_{-\infty}^{\infty}x(t)e^{-jwt}\mathrm{d}t$$ During a lecture a few months ago in my signals and systems class, the professor showed when the CTFT ...
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2answers
175 views

Prove that between any two roots of $f$ there exists at least one root of $g$

$$f(x)= 1 - e^x\sin(x)$$ $$g(x)= 1 + e^x\cos(x)$$ Prove that between any two roots of $f$ there exists at least one root of $g$. I know Rolle's Theorem and the Intermediate Value Theorem (I ...
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3answers
48 views

Can you help me prove that this function increases? [closed]

Can you help me prove that $f(x) = x/\ln (\ln x)$ is increasing on $(e^2, +\infty)$?
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1answer
26 views

Proof of the Primitive Existence Theorem

In the first section of the proof of the primitive existence theorem we are trying to show that $$F\left(z+h\right)-F\left(z\right)=\int_{L_{z,z+h}}f$$ where $\alpha, z,z+h$ are collinear and ...
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3answers
38 views

Use direct proof to prove: If $A \cap B = A \cap C$ and $A \cup B = A \cup C$, then $B = C$

I'm interested in knowing if the method I used is correct. I've been teaching myself proofs lately and I am having difficulties with how to approach a problem so any general tips would be awesome as ...
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0answers
43 views

The Maximum Modulus Principle Applied to the Proof of Schwarz Lemma

I am using the following statement of the Maximum Modulus Principle: Theorem: Let $G$ be a region and let $f$ be holomorphic on $G$. Suppose $\exists~ a \in G$ such that $|f(z)| \leq |f(a)| ~ ...
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2answers
41 views

Flawed proof that the closure of a set is closed?

So I'm reading baby Rudin's third edition and on page 35, he shows a proof that the closure of a set is closed. (I'm not questioning the result, but it seems to me the proof has a mistake and I'm ...
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1answer
6 views

Fourier transform integral for even function proof

Using fourier transform prove that: If $f(t)$ is even $$F(v)= 2\int_0^\infty f(t)cos(2\pi\nu t) \,\mathrm dt$$ How to do that?
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15 views

Correctness of use induction in the proof

"Let $S$ be a subset of vector space $V$. Let $P_1, ... , P_n$ be elements of vector space $V$. Let $S$ be the set of all linear combinations $t_1 P_1 + ... t_n P_n$, with $0 \le t_i$ and $t_1 + ... ...
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1answer
23 views

Proof of a Property of a Monoid: Show that the following are equivalent

I am stuck on the following question: Show that the following are equivalent for a monoid M: If $ab$ is a unit, then both $a$ and $b$ are units If $ab = 1$, then $ba = 1$ I am able to show that ...
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1answer
40 views

Strong induction postage stamp problem [on hold]

Using 5 cent, 11 cent, and 17 cent stamps, what are all possible amounts of postage that can not be formed? Prove your answer. Part of your answer needs to use strong induction.
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1answer
42 views

Proving binomial identities [duplicate]

Can someone help me prove these two binomial identities using either walks in Pascal's triangle or a committee-selection model? $(1)$ $\qquad$ $\displaystyle\sum_{k=0}^m {m\choose k}{n\choose ...
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2answers
25 views

Understanding a proof about permutations from P.A.Grillet's “Abstract Algebra”

I need a hand in understanding the following proof of the following theorem(by P.A.Grillet in his textbook "Abstract Algebra"). Proposition $4.1$. Every permutation of $\{1,...,n \}$ is a product ...
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1answer
40 views

Proof of indexed family set

I am stuck at the following exercise in Velleman's How To Prove it: Suppose $\{A_i\mid i \in I\}$ is a family of sets. Prove that if $\mathscr P\left(\bigcup_{i\in I}A_i\right) \subseteq ...
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0answers
26 views

Relation between chain rule and implicit differentiation derivation in multi variable calculus

So my question is on the derivation of the implicit differentiation (taken from here). The general chain rule, from here, it says that if we have a function $z$ of $n$ variables, $x_1, ...
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1answer
54 views

In the proof of the existence of $n^{\text{th}}$ roots (Rudin, Theorem 1.21), why is $y-k$ an upper bound of $E$?

After defining $E = \{ t > 0 : t^n < x\}$ and $y = \sup E$, Rudin first proves that the assumption $y^n < x$ leads to a contradiction. Then he continues Assume $y^n > x$. Put $$k = ...
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0answers
17 views

The Cramer-Rao Lower Bound proof

Let $X_1, . . . , X_n$ be i.i.d. with density function $f (x|θ)$. Let $T = t (X_1, . . . , X_n)$ be an unbiased estimate of $θ$. Then, under smoothness assumptions on $f (x|θ)$, $$Var(T) >= ...
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3answers
23 views

Question about the necessary condition for disjoint set

I ran into a problem asking about the necessary condition for disjoint sets and am quite puzzled by the solution to it. It is given as follows: Let $A$ and $B$ be subsets of a universal set. Which ...
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1answer
38 views

Intermediate value for derivative, Apostol text

In the textbook, Mathematical Analysis of Apostol, there is the intermediate value theorem for derivative as shown below, Now after the theorem there is the note that this theorem is also true if ...
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3answers
57 views

In proving A = B, A, B are sets, do you always have to show $\subseteq$ and $\supseteq$?

I am trying to show the DeMorgan's Law $X \backslash \bigcup_{\alpha \in I} A_\alpha = \bigcap_{\alpha \in I} (X \backslash A_\alpha)$ It seems I could directly approach this as follows: $X ...
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5answers
88 views

Showing that $n! > n^2$ for $n\geq4$ by induction

My attempt: Prove $ n! > n^2 $ for $ n \geq 4 $ Base Case: $P(4) = 24 > 16$ Inductive Hypothesis $P(k) : k! > k^2 $ $P(k+1) : (k+1)! > (k+1)^2 $ $ (k + 1)! - (k+1)^2 > 0 $ $ ...
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1answer
51 views

Great Circles in $SU_{2}$

So I am working on the proof that all great circles in $SU_{2}$ (circles of radius 1) are a coset of a longitude, and I am unsure what a great circle looks like in matrix form. Clearly any point on ...
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0answers
24 views

Proving that for every $\lambda$ there exists an $x \in \mathbb R$ so that $x(p(x)-2)^2=\lambda$, where p(x) is a non-constant polynomial.

The question is: Prove that for every $\lambda$ exists an $x$ such that $x(p(x)-2)^2=\lambda$, where $p(x)$ is a non-constant polynomial, and $\lambda \in \mathbb R$ So I've gone ahead and opened up ...
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0answers
8 views

Why $x_B=\tilde b +\tilde A x_{\bar B};c^Tx=\psi+\tilde c^Tx_{\bar B}$ doesn't describe an optimal solution iif $\tilde c_i\le 0,\forall i$

How to counterprove the assertion that if a feasible dictionnary in the type \begin{cases} x_B=\tilde b +\tilde A x_{\bar B}\\c^Tx=\psi+\tilde c^Tx_{\bar B} \end{cases} describe an ...
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1answer
27 views

Understanding shift to polar coordinates in the newtonian central force system of ODE's

This is from Hirsch, Smale and Devaney chapter 13. The larger context is moving towards blowing up the singularity at the origin of the system. The second order ODE is defined, $X:t\rightarrow ...
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23 views

Show that if all row-sums of a square matrix $A$ are equal to $0$, then $A$ is singular [duplicate]

I need to show that if all row-sums of a square matrix $A$ are equal to $0$, then the matrix is singular. My idea was that to represent the situation, I can do as follows: $$A\vec{x} = \vec{0}$$ ...
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1answer
23 views

Proof for $\forall x \in R^+, x^2 + y^2 + z^2 \geq xy + xz + yz$

I am doing a question to practice doing proofs with real numbers as I am still not so good at it. I ran into some problems for the following question where it is as given: $\forall x \in R^+, x^2 + ...
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1answer
29 views

If $X$ is compact then a subset of $C(X)$ is compact if and only the subset is closed, uniformly bounded and equicontinuous

I'm having trouble understanding a part of this following proof. The theorem is that is $X$ is a compact metric space and $\mathcal F \subset C(X)$ then $\mathcal F$ is compact if and only if ...
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19 views

How can I verify the following equality?

$$\int_0^{\infty}\frac{C\exp(-\frac{mx^2}{\Omega})}{\Omega^m}\frac{1}{\sqrt{2\pi}\lambda\Omega}\exp\left(-\frac{(\ln ...
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2answers
22 views

Confused about a neither statement and modular

I am trying currently in the process of learning proofs involving congruence of integers with methods of direct and contrapositive and proofs with cases. However, I am quite confused by this statement ...