# Tagged Questions

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

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### Proof regarding little o for vector field

I'm struggling with one part of a proof of a theorem. Let $\gamma(t) : A \subseteq \mathbb{R} \rightarrow \mathbb{R}^m$, with $\gamma \in \mathscr{C}^1(A)$ (hence $\gamma$ differentiable). ...
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### Alternative proof: show that any metrizable space $X$ is normal - Part 2

This is a follow up to one of my earlier questions I am reading some stuff online and saw a proof as follows Per a comment in Part 1 in linked, We know that $d(C_1,C_2)$ could easily be zero ...
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### Using Rolle's Theorem to prove that f'(c)=rf(c). [on hold]

Suppose that the function $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a)=f(b)=0$. Prove that for each real number $r$, there is some $c$ on $(a,b)$ such that $f'(c)=r\cdot f(c)$. ...
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### proof of preimage of union and intersection of sets

I was learning to proof the following proposition "the inverse image of an intersection or union equals the intersection or union of the inverse image" following these two really good youtube videos: ...
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### Discrete Math Understanding a proof involving the definition of divisibility

In this first course on discrete mathematics, the instructor provided this following solution to a question. The question was asked us to prove the following (the solution is provided as well): My ...
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### The intuition behind choosing this length?

In Problem 1, RMO 2004 there is a particular choice of length which leads to the solution, the length being that of the tangent from the foot of the perpendicular to the circle. Just a rundown of ...
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### Discrete Math Proof verification: products of floor

Determine if the following is true or false and provide a proof: $\forall x\in\mathbb{R},\exists y\in\mathbb{R}$ so that $\lfloor xy\rfloor = \lfloor x\rfloor \lfloor y\rfloor$ My attempt: -The ...
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### Class equation and orbit stabilizer theorem

I was reading the proof of the following theorem but I cannot understand how to use the class equation as he wants me to. Theorem Suppose that $G=HK$ where $H$ is a normal locally finite $p'$-...
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### Law of total expectation?

Apparently $E[X] = E[E[X\mid Y]]$ but I don't understand what this really means. I looked at https://en.wikipedia.org/wiki/Law_of_total_expectation but need another explanation. Isn't this the same ...
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### Proving $\text{Var}(X) = E[X^2] - (E[X])^2$

I want to understand something about the derivation of $\text{Var}(X) = E[X^2] - (E[X])^2$ Variance is defined as the expected squared difference between a random variable and the mean (expected ...
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### An unexplained condition on $a$ in a proof on the primes?

Lemma A positive integer $n$ is a prime if $(n,p) = 1$ for every prime integer $p \leq \sqrt{n}$ Proof in my text Let $(n,p) = 1$ for every prime $p \leq \sqrt{n} \:$. Suppose $n$ is not a ...
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### Why is $<T\vec x,\vec y>=<\vec x,T^*\vec y>$ for hermitian matrix T?

In this video in 5:15 there's a proof that every hermitian matrix has real eigenvalues. I don't understand the step: $<\vec x, L\vec y>=<L^*\vec x,\vec y>$. I know that I can pull out ...
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### Proof for $|z_1+z_2| \le |z_1|+|z_2|$ and $|z_1-z_2|\ge |z_1|-|z_2|$ [closed]

I need proof for $$|z_1+z_2| \le |z_1|+|z_2|$$ and $$|z_1-z_2|\ge |z_1|-|z_2|$$
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### How to prove this using laws? [closed]

How do they prove this? $$(p\to q)\land[\neg q\land(r\lor\neg q)]\equiv\neg q\land\neg p$$
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### Need help with proof about Diophantine equations

The way I am planning to arrange this is by providing fragments of the proof, so I can understand what's going on before forging ahead, so if you are going to help me, keep in mind that I am going to ...
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### Disproving existence statements

I am trying to get some practice on disproving existence statements and I was really stuck on this one: "There exists an example of three distinct positive integers different from $a,2a,3a$ for some ...
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### Question about the proof of the fact that minimum is attained for a l.s.c. convex function over a convex compact set.

I quote here the proof of a result given in Haim Brezis Functional Analysis, Sobolev Spaces and partial differential Equations: I haven't been able to conclude where exactly is this hypothesis used: ...
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### Question about the proof $X'$ reflexive $\Rightarrow X$ reflexive.

I have a doubt in the proof I have been given of the fact: For a Banach space $X$, if $X'$ is reflexive then $X$ is reflexive. This is proven by showing first theorem 1 and theorem 2, which I quote ...
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### Proving the Fermat's theorem

Fermat's theorem: if a is not divisible by p, then $a^{p-1} \equiv 1 \pmod p$ Since $\varphi(p)=p-1$, this is a special case of Euler's theorem. If $(a,m)=1$, then $a^{\varphi(m)}\equiv 1 \pmod m$. ...
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### Question about Brouwer degree under uniform convergence.

I was wondering the following: Say a smooth sequence $u_k$ on a smooth manifold converges uniformly to the limit $u$. Does $u$ preserve the Brouwer degree of the $u_k$'s? I also believe this is an ...
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### Not understanding the proof that there is no surjection from a set to its powerset

Here is the question: If a set, $A$, is finite, then $|A| < 2^{|A|} = |P(A)|$, and so there is no surjection from set $A$ to its powerset. Show that this is still true if $A$ is infinite. ...
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### Movie math, real or not? (Ghost busters trailer) [closed]

I was watching the ghost busters trailer, and I stopped on a frame containing 'quantum physics'. I wonder if all of this is meaningful math, or some of it is complete gibberish, to look smart. I see ...
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### Calculating eigenvalues of an infinite sequence

Consider the set V of all infinite sequences in $\Bbb{F}$. $$V = { \{ (a_0, a_1, a_2, ...)\ |\ a_i \in \Bbb{F}, \text{for } i=0, 1, 2, ... \} }$$ and the function T:V $\rightarrow$ V defined by T(...
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### Given a triangle with a segment inside it parallel to a segment of the triangle, prove that one of the sides times another equals a third (picture).

Explained in picture below. I don't understand the relevance of the fact that AD = 1. Thanks.
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### How can you prove a quadrilateral given a diagonal and segments going from the vertices to the diagonal (picture)?

I'm having quite a bit of trouble with this proof. The angles formed by the segments between diagonals and vertices are 90 degrees, and the vertices on the diagonal and the diagonals' vertices are ...
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### Why the set of points satisfying (2x-x²-y²)*(x²+y²-x)>0 is the same of the set with condition ((2x-x²-y²)>o and (x²+y²-x)>0)?

Why the set of points on $\Bbb R^{2}$ satisfying $(2x-x²-y²)(x²+y²-x)>0$ is the same of the set of all elements on $\Bbb R^{2}$ with condition $((2x-x²-y²)>0$ and $(x²+y²-x)>0)$, i think ...
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### Formal proof on continuity of multivariable function.

Let $B(o,r) =${$(x,y)\in \Bbb R: \left\lVert (x,y) \right\rVert<r$} , to some r>0 and the norm is an euclidean norm, let $f(x,y)$ -> $L$ as (x,y) -> (0,0), with f : $B(o,r)$ -> $\Bbb R$. ...
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### Intuition behind the proof of the validity of the Euclidean algorithm

As the question title suggests, could anybody explain to me their intuition behind the proof of the validity of the Euclidean algorithm?
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### Need help in understanding some basic ordinal concepts

I'm trying to construct a proof that there is no homeomorphism from a subspace of $\omega_1$ to $\Bbb{Q}$ with the usual topology, but first I need to clear up some concepts. This is the definition ...
I've got a question about a descending sequence of closed sets. Milnor writes in his book "FROM THE DIFFERENTIABLE VIEWPOINT". In his proof of Sards Theorem he wrote: Let $f:U\rightarrow \mathbb{R}^p$ ...