For questions concerning a specific proof, asking for verification, identifying errors, suggestions for improvement, etc.

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Spivak Ch1 Proof Critiques

I've started working through Spivak's Calculus. I'm going into senior year after this summer, took the AP Calculus BC test last year, and wanted to get a firmer foundation in calculus before I take ...
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1answer
15 views

$\lbrace \lim f_n(x) \rbrace$ is a Borel set if each $f_n$ is borel

Suppose for all $n$ that $f_n:\mathbb{R}\to \mathbb{R}$ is Borel measurable. What follows is an attempt of the proof that $\lbrace x: \lim_{n\to \infty} f_n\rbrace$ is Borel measurable, but I am a bit ...
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2answers
65 views

Proof that if $a,b \in G$ and $a^4b = ba$ and $a^3 = e$ then $ab = ba$

I tried to prove one of the examples in my Abstract Algebra book that stated: Prove that if $a,b \in G$ and $a^4b = ba$ and $a^3 = e$ then $ab = ba$ I went about just saying that $a^4b = ba ...
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2answers
39 views

Prove that if $Z(G) = \lbrace x \in G: gx = xg \text{ for all }g\in G\rbrace$ then $Z(G)$ is a group

So my challenge is: Prove that if $Z(G) = \lbrace x \in G: gx = xg \text{ for all }g\in G\rbrace$ where $G$ is a group, then $Z(G)$ is a group Unlike this question: To show that the center is a ...
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1answer
21 views

Characterization of subsets of $\mathbb{R}^n$ of the form $X+Y$

The following comes from the mathematical tripos exam at Cambridge: Let $X,Y \subset \mathbb{R}^n$, and define $X+Y = \{x+y : x \in X, y \in Y\}$ Prove or disprove each of the following: (i) If ...
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39 views

Proof that odd perfect numbers cannot consist of single unique factors?

I'm a high school student, so please point out my mistakes nicely :) So we already know odd perfect numbers cannot be in the form of a square, but how about that they cannot be in this form: ...
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0answers
29 views

How to check whether the following function is concave or convex or neither.?

Let $\pi$ be a vector such that all its elements sum to 1. i.e, $\sum_1^n \pi(i) = 1$ where $\pi(i)$ denotes the i$^{th}$ component and $n$ is the length of the vector. Let $D$ be a diagonal matrix ...
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1answer
40 views

Understanding a part of a proof involving Hilbert-Schmidt norm

I came across a proof I do not seem to understand fully, a screenshot is provided below. my concerns are the following: Why does the fact that $||T||_2 = ||UT||_2$ for every unitary U, allow us ...
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2answers
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Proving that $\sum_{i=2}^n(5i-4)=\frac{n(5n-3)-2}{2}$ for all $n\geq 1$ by mathematical induction

I have this question: Show, using mathematical induction, that for all natural numbers $n$, $$6 + 11 + 16 + 21 + \cdots + (5n-4) = \frac{n(5n-3)-2}{2}$$ I am confused in that that question states ...
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1answer
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Prove that if $f$ is an invertible function and $g$ is an inverse, then the codomain of $g$ is equal to the domain of $f$ and vice versa

I am trying to show, without using the bijection properties, what is above. Assume $f$ is an invertible function and $g$ is an inverse of $f$. For $f \circ g $ to be well defined then the image of ...
2
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2answers
41 views

Proof that $n\Bbb Z \leq \Bbb Z$ and are the only subgroups of $\Bbb Z$

My challenge is Prove that if $n = 0,1,2,\ldots$ and $n\Bbb Z = \lbrace nk: k \in \Bbb Z \rbrace$, show that $n\Bbb Z$ is a subgroup of $\Bbb Z$ and are the only subgroups. I handled the first ...
4
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1answer
27 views

Multiplying two inequalities

Suppose we have two inequalities $$a\leq x\leq b\tag{1}$$ $$c\leq y\leq d\tag{2},$$ where $a,b,c,d>0$. Then can I conclude that $$ac\leq xy\leq bd\quad ?$$ My attempt: Since $a,b,c,d>0$ and ...
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3answers
44 views

The equation of a tangent to a circle at a given point

18. Show that the equation of the tangent $PT$ at the point $P \left(\frac{1}{5}, \frac{3}{5}\right)$ on the circle $$x^{2} + y^{2} + 8x + 10y - 8 = 0$$ is $3x + 4y - 3 = 0$. Find ...
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2answers
14 views

Proof of the right and left cancellation laws for Groups

I was asked to proof the right and left cancellation laws for groups, i.e. If $a,b,c \in G$ where $G$ is a group, show that $ba = ca \implies b=c $ and $ab = ac \implies b = c$ For the first ...
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42 views

Ito's lemma - mistake in text book?

Ito gives us $$dW = \dfrac{dW}{dX} dX + \left(\frac{dW}{dt} + \frac{1}{2} \frac{d^2W}{dX^2}\right) \, dt$$ We have a function $W(t) = 1 + t + E^{X(t)}$. My text book says that $$dW = e^{X(t)} \, dX + ...
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1answer
12 views

Proof that an archimedean absolute value on $\mathbb Q$ is equivalent to $|\ |_\infty$

I have included a bolded comment in a step in the part of Gouvea's proof of Ostrowski's theorem where he shows that an archimedean absolute value on $\mathbb Q$ is equivalent to $|\ |_\infty$ (the ...
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5answers
63 views

Show $\sum n e^{-na}$ converges for $a>0$

Is there any test or property in particular I can use to show $ \sum n e^{-n a}$ is convergent for $a>0$ ? I think it is obvious that from looking at the function that this is convergent, since ...
2
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1answer
57 views

Showing Uniform convergence of $\frac{n x}{1 + n \sin(x)}$

I want to prove for all $a\in \left(0,\frac{\pi}{2}\right]$, $ \ f_n\to f$ uniformly on $\left[a,\frac{\pi}{2}\right]$. Also, how is this different from $f_n \to f$ uniformly on $\left(0, ...
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0answers
25 views

normal squared characteristic function derivation

I'm trying to derive the normal squared characteristic function, there's already a question on this but the answer has a part which is "proved as an excercise" which I try to do here. Is my proof ...
2
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1answer
23 views

Proof that there exists an $x \in G$ such that $xa = b$

So this is my challenge: Let $G$ be a group and $a,b \in G$. Then $xa = b$ has a unique solution I went about saying that $xa = b \iff xaa^{-1} = ba^{-1} \iff x = ba^{-1}$. $ba^{-1} \in G$ ...
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32 views

Why is this class closed under difference?

We have two independent random variables $X\perp Y$ involving three spaces: $(\Omega,\mathcal{A},P), (E,\mathcal{E}), (F,\mathcal{F}).$: $$X:\Omega \rightarrow E,\ Y:\Omega\rightarrow F$$ My book says ...
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22 views

Isometry algebra implication from Riemannian covering

I really wish that, if $\pi:(M,\mathrm{g})\twoheadrightarrow(N,\mathrm{h})$ is a Riemannian covering, then $\mathfrak{i}(N,\mathrm{h})\leq\mathfrak{i}(M,\mathrm{g})$, where ...
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Definition of normal sets and compactness

I am struggling a little bit with this notion. In Conway's Functions of One Complex Variable, he offers the definition: A set $\mathscr F \subset C(G,\Omega)$ is "normal" if each sequence in ...
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1answer
266 views

How many different ways are there to organize 2 groups?

In how many different ways can you split 10 people into two groups with the same amount of people? My attempt: Since the order in which you choose someone doesn't matter, I chose to calculate the ...
0
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1answer
19 views

Give an L-formula ϕ4(x) that defines the interval [1,√2) ⊂ R in M.

Let L = {+, · P} where + and · are binary function symbols and P is a unary predicate symbol, and let M be an L-structure where its domain |M| is the set R of real numbers, + and · are the usual ...
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35 views

Continuous function on compact subset of $\mathbb R$ to itself has a fixed point.

Let $f:[a,b] \to [a,b]$ be continuous. Then $f$ has at least a fixed point. I read the following proof from Limaye book. Define $F(x)=f(x)-x.$ Since $a \leq f(x) \leq b,\ \quad F(a)\leq 0 \ \quad ...
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1answer
64 views

Can the expression $6^{2n} - 25$ be a prime for all $n \geq 2$?

Can the expression $6^{2n} - 25$ be a prime for any $n \geq 2$? My attempt to solve the problem: No, it cannot. $6^{2n} - 25 = (6^{n})^{2} - 25 = (6^{n})^{2} - 5^{2} = (6^{n} + 5)(6^{n} - 5)$ And ...
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3answers
71 views

For a function Y : X→X , if Y is injective, then Y∘Y∘Y is injective.

For a function Y : X→X , if Y is injective, then Y∘Y∘Y is injective. My attempt: Using contrapositive, if Y is not injective. then Y ∘ Y is not injective, the there exist x, x' ∈ X with x ≠ x' but ...
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1answer
34 views

Cases of $A^2 = -I$. Why is there a contradiction when reusing this proof?

I had to prove that $\nexists ~A \in M_{3,3}(\mathbb R) : A^2 = - \mathbb I.$ I argued $$\iff A=-A^{-1}$$ $$\iff \det( A)=\det(-A^{-1})$$ $$\iff \det( A)=(-1)^n\det A^{-1}$$ $$\iff \det (A) + \det ...
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2answers
61 views

Let S be the set of stars in our galaxy. There is same number subsets as functions f : S --> {a,b}? [duplicate]

Question: Let $S$ be the set of stars in our galaxy. There is exactly the same number of subsets of stars in our galaxy as there are functions $f:S \to \{a,b\}$ . My solution is; True.However, I'm ...
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1answer
55 views

Mathematical Induction. Horses made me question my understanding

I recently read about the false inductive proof that all horses are the same colour. There are some mathSE threads about this already (MathSE_thread_1, MathSE_thread_2). After reading this, I now ...
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1answer
31 views

If $q^k n^2$ is an odd perfect number with Euler factor $q^k$, can $q = 73$ hold?

Call a number $N$ perfect if $\sigma(N)=2N$ where $\sigma$ is the classical sum-of-divisors function. If $N = q^k n^2$ is an odd perfect number, can $q = 73$ hold? Here is my attempt: Since ...
3
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0answers
37 views

Order of prime ideals over split primes in the class group.

Let $P$ be a prime ideal of $\mathcal O_K$ ($K$ a quadratic field) and let $P$ have norm $p$ where $p$ is a split prime. Is it possible for the ideal class $[P]$ to have order less than three? I feel ...
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28 views

Does $[E:F]=|Aut(E/F)|$ imply Galois extension?

Let $E/F$ be a finite field extension such that $[E:F]=|Aut(E/F)|$. Then, is $E/F$ Galois? Even though I have proven it, I'm not sure of it. Is this really true? Here's how I proved it: Let $\bar ...
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5answers
108 views

There exist a set $X$ such that the number of function $y: x\to \{1,2,3\}$ is $1000$.

There exist a set $X$ such that the number of function $y: x\to \{1,2,3\}$ is $1000.$ My attempt: False, Let set $z = \{1,2,3\}$ then $|z|^{|x|}$ is set of function $y:x\to z.$ $|x| = n$ and $|z| = ...
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Is there any mistake? Proof related to the Poisson summation formula.

I need to prove the following statement: Let $\varphi \in C(\Bbb R)$ with compact support. Then, $$ \Big \Vert \sum_{k\in \Bbb Z} \varphi(k) e^{ikx} \Big \Vert_{L_1(0,2\pi)} \leq C \Vert ...
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40 views

Show $\lim_{n\to\infty}(Lu_n) = L(\lim_{n\to\infty} u_n)$

Suppose $\{u_n\}$ is a convergent sequence in Hilbert space $H$ and $L$ is a bounded (continuous) linear operator on $H$. Use the definition of convergence to show that $\lim_{n\to\infty}(Lu_n) = ...
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1answer
39 views

Are the extrema of this function global or local?

Last question about this function, I promise. The function $f: \mathbb R \rightarrow \mathbb R$ is given by $$f(x) = \begin{cases} \frac{x^2+5x+7}{x+3} & \mathrm{for} \; x < -3 \\ 0 & ...
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1answer
26 views

Is this proof correct for $p_1\times p_2 \mod k = p_3$?

I am trying to prove that if $\gcd(p_1, k) = \gcd(p_2, k) = 1$, then $$\gcd(r, k) = 1$$ where $r = p_1\times p_2 \mod k$. This fact is essential to guarantee that a unit group $U(n)$ of a group ...
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2answers
30 views

Prove that a function $f:X \to Y $ is injective if and only if $\forall x_1, x_2 \in X$ where $f(x_1) = f(x_2)$ implies that $x_1 = x_2$

Prove that a function $f:X \to Y $ is injective if and only if $\forall x_1, x_2 \in X$ where $f(x_1) = f(x_2)$ implies that $x_1 = x_2$ Taking the contrapositive we get (this is the step I'm a ...
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1answer
33 views

Why this set is the pole set of $z$?

Suppose $V\subset \mathbb P$ a projective variety, $z$ a rational function on $V$ and let's define $J_z=\{F\in k[X_1,\ldots, K_{n+1}\mid \overline Fz\in \Gamma_h(V)\}$ and $S_z$ the polo set of $z$, ...
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2answers
28 views

Continuous and bounded - Check my proof please

Let $f : [0, ∞) → \mathbb{R}$ be continuous such that $\lim_{x→+∞} f(x) = 0$. Prove that $f$ is bounded on $[0, ∞)$ By our hypothesis and the definition of continuity, given $ c \in [0, \infty), ...
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2answers
48 views

Legendre symbol, a theoretical question.

I need to show that if $p$ is a prime number of the form $p=4m+1$, then for any divisor $d$ of $m$: $$\left(\frac{d}{p} \right) = 1$$ where $\left(\frac{d}{p} \right)$ is the Legendre symbol. My ...
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1answer
34 views

Is the function continuous and differentiable at $x=-2$?

The function $f: (-3, \infty)$ is given by $$f(x) = \begin{cases} \frac{x^2+5x+7}{x+3} & \mathrm{for} \; -3 < x < -2 \\ 1 & \mathrm{for} \; x = -2 \\ -x-e^{-x}+e^2-1 & \mathrm{for} ...
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1answer
30 views

Proof of (a step in the proof of) the Law of Large Numbers

Theorem: Let $f:[0,1] \to \mathbb R$ be a measurable function bounded by $c$. Let $U_1,U_2,\ldots,U_n$ be i.i.d. and Uniform$(0,1)$. Then: $$ P \left( \left\lvert ...
1
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1answer
26 views

In $\mathscr{V}$, let $X \subset \mathscr{V}$ be a set of $n$ vectors. $Y \subset X$ contains vectors all scalar multiples, $X$ linearly dependent.

I would just like to verify that my proofs are sound and receive any suggestions on rewording. (If relevant, I am self-studying and haven't done a serious proof in about a year.) $\mathscr{V}$ is a ...
1
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1answer
23 views

Prove $\vdash\forall x(\alpha\to\beta)\to(\exists x\alpha\to\exists x\beta)$

I want to show that: $\vdash\forall x(\alpha\to\beta)\to(\exists x\alpha\to\exists x\beta)$ I started my deduction as follows: $\vdash\forall x(\alpha\to\beta)\to(\forall ...
2
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2answers
55 views

$x^n + y^n = z^n$, $n>1$ To show that $x,y,z$ is greater than $n$

Problem: If $x$,$y$,$z$ and $n>1$ are natural numbers with $$x^n+y^n = z^n$$ then show that x,y and z are all greater then $n$. My approach, from Fermat's Theorem we know that $x^n + y^n = z^n$ ...
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0answers
30 views

Banach Tarski Notation

Okay, I think I have a full notation and the rules of it how to extend the Banach-Tarski Paradox to an abritary number of cutoffs, as I introduced in Another way of extending the Banach-Tarski ...
3
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2answers
61 views

My proof that an n digit number, times an n digit number can be expressed as a 2n digit number

I am very proud to say this is the first time I've actually used maths to endeavour to prove something without it being related to a question from my course! Statement In a base $B$, an $n$ digit ...