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

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6
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3answers
2k views
+50

Proving rigorously the supremum of a set

Suppose $A \subset \mathbb{R} \neq \emptyset$. Let $A = [\,0,2).\,\,$ Prove that $\sup A = 2$ This is my attempt: $A$ is the half open interval $[\,0,2)$ and so all the $x_i \in A$ look like $0 ...
2
votes
1answer
25 views

Determine if n is a prime.

Let $n$ be a positive natural number. You know the following facts about $n$ . Firstly, $n<10^{6}$ . Moreover, not a single integer $k$ between $1$ and $10^{4}$ divides $n$ . Does it ...
1
vote
1answer
21 views

A problem on finding the nearest points to the origin on the intersection of two surfaces

Suppose we are to find the points nearest to the origin on the curve of intersection of the two surfaces $g^{-1}_{1}\{ 0 \}$ and $g_{2}^{-1}\{ 0 \}$, where $g_{1}: (x, y, z) \mapsto x^{2} - xy + y^{2} ...
0
votes
0answers
17 views

formal proof that p-values are uniformly distributed

I'm trying to prove that $p$-values under the null hypothesis are uniformly distributed in $[0, 1]$ for an absolutely continuous test statistic $X$. Proof: By continuity of $F_X$, it is sufficient to ...
5
votes
2answers
32 views

Let $1 \leq p <\infty$ and $f \in L^p(\mathbb{R})$. Prove $\lim_{x \to \infty} \int_x^{x+1} f(t) dt = 0$.

(Jones, p. 246) Let $1 \leq p <\infty$ and $f \in L^p(\mathbb{R})$. Prove $\lim_{x \to \infty} \int_x^{x+1} f(t) dt = 0$. This seems pretty easy to prove in the following way: Let $g_j$ be a ...
1
vote
0answers
182 views

$\pi(z)-\omega(z-1)-\{-1,0,1\}= \pi(2z-1)- \pi(z)$ when $z(z-1)$ is divisible by all primes ${<}\sqrt{z}$

I have encountered the below problem: Given $z(z-1)$ divisible by all primes ${<}\sqrt{z}$ (and the prime factors of $z(z-1)$ are consecutive primes), prove (or disprove) ...
4
votes
1answer
80 views

Existence of bijection that reorders elements?

Suppose I have some function $f:\mathbb{R}\to[0,1]$. Does there necessarily exist a bijective mapping $g:\mathbb{R}\to\mathbb{R}$ such that $g(x)\leq g(y)$ implies $f(x)≤f(y)$? If not, does it help if ...
0
votes
0answers
11 views

Argument for finite solution of power Diophantione Equation.

Assume the equation $4x^3=y^2+3$ has infinite positive integer solution. If $x,y$ has general solution then it is clear that for any $x$(rational, integer), there is a $y$. It can be said there is a ...
3
votes
0answers
135 views

An argument for “Brocard's problem has finite solution”

Brocard's problem is a problem in mathematics that asks to find integer values of n for which $$x^{2}-1=n!$$ http://en.wikipedia.org/wiki/Brocard%27s_problem. According to Brocard's problem ...
2
votes
0answers
49 views

Is $\forall n\exists m:\, m^2=n,\text{ where }m,n ∈ \mathbb N$ true or false?

$\forall n\exists m:\, m^2=n,\text{ where }m,n ∈ \mathbb N$. Prove whether this expression is true or false. My attempt: False, take $n=3,$ then there is no such integer $m$, such that $m^2=3$. Thus, ...
3
votes
1answer
30 views

Prove that square of even integer is even.

Is my proof correct? Suppose $n=2m$ is an even integer. Since $n=2m$ , then $n^{2}=(2m)^{2}$ $n^{2}$ = $(2m)^{2}$ = $4m^{2}$ =$2(2m^{2)}$ Since $(2m^{2})$ is an integer and $2(2m^{2})$ ...
1
vote
1answer
19 views

Deciphering proof of SLLN

I was looking at a proof of the string law of large numbers, and am having trouble finding where the proof uses the assumption that the random variables are identically distributed. I'll reproduce the ...
7
votes
2answers
111 views

$\forall x\in\mathbb{R},,$ if $x^{2}$ is rational, then $x$ is rational.

This is my attempt at this question. Is this correct? $\forall x\in\mathbb{R},,$ if $x^{2}$ is rational, then $x$ is rational. This statement is false. Using counterexample, let $x=\sqrt{2}$. Since ...
2
votes
0answers
29 views

A question regarding Parseval's identity.

In most books/websites, Proposition 2 (see below) is either stated for a Hilbert space or proved via Riesz-Fischer. Does the follow approach (which seems to work in an inner product space) fall down ...
0
votes
1answer
28 views

$f:U \rightarrow \mathbb{R}$, $U$ is an open conected subset of $\mathbb{R}^n$ and $f \in C^1$ need to show that $f$ is $M$ Lipschitz on any compact

It is a more general form of the question here, only here $U$ is not a convex set but an open and connected subset of $\mathbb{R}^n$. I need to show that $f$ is $M$ Lipschitz on any compact $K \subset ...
2
votes
0answers
29 views

Is $f\colon Y'\to Y$ continuous?

Consider $X=\left\{0,1,2\right\}^{\mathbb{Z}}$ and $T\colon X\to X$ continuous, describing the following dynamics: For $\eta\in X$ let $\eta(y)$ describe the y-th position in the bi-infite sequence ...
0
votes
0answers
21 views

Precompact and locally finite implies finite intersection

An exercise in Lee's Introduction to Smooth Manifolds asks the following: Let $M$ be a topological manifold, and let $\mathcal U$ be an open cover. Suppose the sets in $\mathcal U$ are precompact ...
0
votes
0answers
14 views

Let $R$ be a ring with 1 and N be a submodule of R-module M. If $M$ is free of finite rank, is $M/N$ necessarily free of finite rank?

Let $R$ be a ring with 1 and N be a submodule of R-module M. If $M$ is free of finite rank, is $M/N$ necessarily free of finite rank? My idea: No. Consider $R = M = Z_6$ and $N = 2Z_6 = \{2,4\}$. ...
0
votes
2answers
35 views

Prove an eigenvector for two matrices is also the eigenvector for the product of those matrices. [duplicate]

So let's assume that A and B are both nxn matrices, and that u is an eigenvector for both A corresponding to lambda one and B corresponding to lambda 2. I need to prove that u is also the eigenvector ...
1
vote
2answers
41 views

Summation Proof Dealing With 3s Multiples [duplicate]

So the problem is as follows: Prove that if the sum of digits of a decimal $n$ is three's multiple, then n is three's multiple by direct proof. For example, $11234567$ is 3's multiple because ...
2
votes
2answers
37 views

Proof for $0a = 0$

Is this a valid proof for $0a =0$? I am using only Hilbert's axioms of the real numbers (for simplicity). $(a+0)(a+0) = a^2 + 0a + 0a + 0^2 = (a)(a) = a^2$ Assume that $0a$ does not equal zero. ...
0
votes
1answer
341 views

If $A$ is a subset of $C$ and $B$ is a subset of $C$, then the union of $A$ and $B$ is a subset of $C$

If $A$ is a subset of $C$ and $B$ is a subset of $C$, then $A\cup B$ is a subset of $C$. I was considering letting $x$ be an element of $A$ and $B$ and going from there, but I'm not sure that that is ...
0
votes
0answers
11 views

Show that $v(E)=\text{sup}\sum_{j=1}^{n}|\mu(A_j)|$ is a measure.

Background A family $\textbf{X}$ of subsets of $X$ is a $\sigma$ algebra in case: $\phi, \mathbb{R} \in \textbf{X}$ $X \setminus A \in \textbf{X}$ if $A \in \textbf{X}$ If $(A_n) \in \textbf{X}$, ...
-2
votes
0answers
47 views

Summation Direct Proof Help [on hold]

Prove that if the sum of digits of a decimal n is three's multiple, then n is three's multiple by direct proof. For example, 11234567 is 3's multiple because 1+1+2+3+4+5+6+7=24, and in fact, 11234567 ...
1
vote
1answer
59 views

Determine a basis for $\mathbb{Z} \oplus \mathbb{Z}$ which determines a basis for the submodule $N$ generated by $(6,9)$

Proof Clearly the rank of $\mathbb{Z} \oplus \mathbb{Z}$ is $2$, so we must have that the rank of $(6,9)$ is $\leq 2$.Let $e_1 = (1,0)$ and $e_2 = (0,1)$ be a basis for $\mathbb{Z} \oplus ...
0
votes
0answers
25 views

Determine all $n \in \mathbb{N}$ such that $GCD(n,48)=6$, $14|n$ and $|Div^+(n)|=12$.

Determine all $n \in \mathbb{N}$ such that $\gcd(n,48)=6$, $14|n$ and $|Div^+(n)|=12$. What I did: $14|n$ then $2|n$ and $7|n$ so $n=2\cdot7\cdot q$, $q \in \mathbb{Z}$. Then $6|n$ implies $2|n$ and ...
0
votes
1answer
57 views

dimensions of two subspaces of a vector space not equal

I have a problem to find a relationship between two subspaces of a vector space. The two subspaces are $W_1$ which is the span of {$v_1,v_2,...,v_{n-1}$} and $W_2$ which is the span of ...
2
votes
1answer
44 views

Proving $(p\to q)\land(p\to r) \equiv p\to(q\land r)$ using logic laws — short cut or incorrect?

Working through this problem: Using logic laws, show that the following are logically equivalent: $$(p\to q)\land(p\to r)\qquad\text{and}\qquad p\to(q\land r).$$ The way I did the problem is ...
0
votes
0answers
27 views

Every primitive of an odd function is even (proof)

I'd like to prove that every primitive of an odd function is even. This is my reasoning; FACT: 1: if f(x) is even, then f'(x) is odd [easy to prove]; 2: if f(x) is odd, then f'(x) is even ...
2
votes
1answer
18 views

Discriminant of n algebraic numbers equals $0$ iff the algebraic numbers linearly dependent

Let $K \subset L$ be two number fields with $[L:K] = n$. Let $\{\alpha_i:1 \leq i \leq n\} \subset L$. Then $\operatorname{disc}(\alpha_1 \dots \alpha_n) = 0 \iff \alpha_i$ are linearly dependent ...
0
votes
1answer
80 views

Covariance inequality for $n$ exchangeable random variables

Let $n \in \mathbb{N}$, $n \geq 2$, assume that $X_1,\ldots, X_n$ are exchangeable, square integrable random variables with $\mathbf{E}\bigl[X^2_1\bigr] < \infty$. Prove that the following ...
0
votes
0answers
23 views

I'm having troubles to find this parametrization.

I'm reading the Reid's Undergraduate Algebraic Geometry book of algebraic geometry for undergraduates and I have two questions about a proof of an example on the page 19: Red question: Reid said ...
0
votes
1answer
14 views

Inversions and Multiplicativity of the Sign of a Permutation

The question is mainly about showing, for two permutations $\sigma, \pi \in S_{n}$, that $\mathrm{sgn}(\sigma \pi) = \mathrm{sgn}(\sigma) \mathrm{sgn}(\pi)$ using inversions of permutations (i.e. a ...
2
votes
0answers
28 views

Determine whether this series converges (proof verification)

Determine whether the following series converges: $$\sum_{n=2}^\infty\frac{(-1)^n}{\sqrt{n}-(-1)^{[\sqrt{n}]}}$$ where $$[x]=\max\{k\in\mathbb{Z}: k\leq x\}$$ My attempt: First I write ...
0
votes
3answers
39 views

Find $GCD(n^2+1,n+1)$

$GCD(n^2+1,n+1)$, $n\in \mathbb{N}$ What I did: $n^2+1=(n-1)(n+1) + 0$ So I thought $(n^2+1:n+1)=n+1$ But that doesn't seem to be the case: $n=2$ $n^2+1=5$ $n+1=3$ $GCD(5,3)=1$ Why is the ...
3
votes
0answers
46 views

Proving a strictly decreasing sequence which tends to zero is positive

Suppose $(a_n)$ is a strictly decreasing sequence such that $a_n\underset{n\to\infty}{\rightarrow}0$. I'm asked to prove that $(a_n)$ is positive. My approach: suppose there is a negative element ...
1
vote
0answers
22 views

Show that the fix points of a function couldn't be in the interior

I want to solve the following problem: Show that the fix points of a function $f:\mathbb B^n\rightarrow \mathbb B^n$ could possibly not be in the interior. By this, Show that the Brouwer fixed-point ...
1
vote
1answer
23 views

I've proved everything about the ideal correspondence easily except $\pi ^{-1} \pi (\frak{a}) = \frak{a}$

The correspondence theorem to which I refer is the bijection between ideals of a commutative ring with $1$, $A$, and ideals of $A/\frak{b}$. I can prove easily most parts that imply the bijection ...
1
vote
2answers
52 views

If $P(A) < P(A \cup B)$, does that mean that $A\subsetneq (A\cup B)$?

If $P(A) < P(A \cup B)$, does that mean that $A\subsetneq (A\cup B)$? I thought that by monotonicity, which states that if $A \subseteq B$ then $P(A) \le P(B)$, then: If $P(A) < P(A \cup ...
3
votes
2answers
59 views

Prove that $x^3 + y^2$ is irreducible in $\mathbb{Q}[x,y]$.

Prove that $x^3 + y^2$ is irreducible in $\mathbb{Q}[x,y]$. My proof: $\mathbb{Q}[x,y] = \mathbb{Q}[x][y]$. Suppose $x^3 + y^2$ is reducible. Then $x^3 + y^2 = (y + g(x))(y + h(x)) = y^2(1 + h(x) + ...
0
votes
0answers
14 views

If $\lim \limits_{x,y\to \infty}f(x,y)=l$ then $\lim \limits_{x,y\to \infty}|f(x,y)|=|l|$

Let $f:\mathbb R^2\to \mathbb R$ if $\lim \limits_{x,y\to \infty}f(x,y)=l$ then $\lim \limits_{x,y\to \infty}|f(x,y)|=|l|$ My attempt: Let $\epsilon>0$, we know that $\exists M>0$ such that ...
4
votes
5answers
254 views

Show that $g(a) = g(b) = 0,\ \int_a^b f(x)g(x)dx=0 $ implies $f(x)=0$

Suppose $f$ is continuous on $[a, b]$, if for every continuous function $g$ on $[a, b]$ with $g(a) = g(b) = 0, \int_{a}^{b}f(x)g(x) dx = 0$, Show $f(x) = 0, \forall x \in [a, b]$, I want to ...
0
votes
0answers
54 views

Prove that $\int_{a}^{b} f(x)g'(x) dx = 0$ iff $f$ is constant

Given that $f$ is continuously differentiable and increasing on $[a, b]$, $g$ is differentiable on $[a, b]$, and $g'$ integrable on $[a, b]$. If $g$ is positive and $g(a) = g(b) = 0$, show that ...
2
votes
0answers
27 views

Generalization of Dirichlet convolution

The Wikipedia page on the Mobius inversion formula gives the following formula in passing: if $$G(x)=\sum_{k=1}^x \alpha(x)F(x/k)$$ for some arithmetic function $\alpha(n)$ possessing a Dirichlet ...
0
votes
2answers
588 views

Proving the Heisenberg Group is a group

I have to prove that the Heisenberg Group, \begin{pmatrix} 1&a&b \\ 0&1&c\\ 0&0&1 \end{pmatrix} where $a,b,c\in\mathbb{R}$ is a group. I am proving a group ...
4
votes
3answers
302 views

Prove that $\sqrt[3]{5-\sqrt{2}}$ is not a rational number

Prove that $\sqrt[3]{5-\sqrt{2}}$ is not a rational number My attempt: Consider the polynomial $ (x^3-5)^2 - 2 = x^6 -10x^3 + 23 = 0 $. By the rational root theorem, we can conclude that $ \pm ...
2
votes
2answers
48 views

Proof that $-(-x) = x$ using just the field axioms

This is my attempt based on some stuff I have been seeing around: Let $y = -x$, then $-y = -(-x)$. Now, lets sum $y + x = (-x) + x = 0$, then we have $y + x = 0$. If we had the additive inverse of ...
0
votes
0answers
19 views

Showing that the 2-torus is parallelizable

Here is the question Let $$ \widehat{\xi}: \mathbb{R}^2 \to \mathbb{R}^2 $$ be a smooth function satisfying $$ \widehat{\xi}(x,y)=\widehat{\xi}(x+m, y+n) $$ for all $x,y\in \mathbb{R}, ...
2
votes
0answers
60 views

Proof that $1 > 0$ using the field and order axioms

This is the problem that you see the first time, and you say: this is the easiest math problem, but you are not quite correct. My reasoning is based on the Peano axioms also. Basically, if 2 numbers ...
3
votes
5answers
140 views

Prove that $2^n\le n!$ for all $n \in \mathbb{N},n\ge4$

The problem i have is: Prove that $2^n\le n!$ for all $n \in \mathbb{N},n\ge4$ Ive been trying to use different examples of similar problems like at: ...