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

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3
votes
1answer
13 views

Proving something is not a Normal Extension

Let $M = \mathbb{Q}(\sqrt{3}, i\sqrt[4]{5})$ be an extension of $\mathbb{Q}$. Then work out the basis of $M$ over $\mathbb{Q}$ and show that the extension $M/\mathbb{Q}$ is not a normal extension. So ...
0
votes
2answers
17 views

Let $p_n$ denote the $n$th prime and $c_n$ the $n$th composite number. Determine all positive integers $n$ such that $|p_n − c_n| = 1$.

The only ones with the property in title I could find are $n = 5$ and $n = 6$: $|11 − 10| = 1$ and $|13 − 12| = 1$. Past the $20$th prime, the list of primes grows too fast for composite numbers. Is ...
8
votes
1answer
68 views

If $\lim_{n\to\infty} (a_{n+1}-a_n)=0$ and $|a_{n+2}-a_n|<\frac{1}{2^n}$ then $(a_n)$ converges

Let $(a_n)$ be a sequence such that $\lim_{n\to\infty} (a_{n+1}-a_n)=0$ and $|a_{n+2}-a_n|<\frac{1}{2^n}$ for all $n$. I have to decide whether or not $(a_n)$ converges. My attempt: I think it ...
2
votes
2answers
29 views

Give a formal argument why there is no planar graph with ten edges and seven regions. Here is my answer

I'm really new to a graph theory, and I have to answer the following question: give a formal argument why there is no planar graph with ten edges and seven regions. Here is my answer: Using ...
0
votes
0answers
22 views

proof that Riemann integrals is extended by Lebesgue integrals

After reading a proof sketch somewhere (forgot the link) I've written a proof in my own words. I'm not quite sure if I got the details right, since there were variants of this floating around that any ...
0
votes
0answers
24 views

Let $R$ be an Euclidean ring. Prove that $\operatorname{lcm}(a,b) = \dfrac{ab}{\gcd(a,b)}$ for $a,b \in R.$

Let $R$ be an Euclidean ring. Prove that $\operatorname{lcm}(a,b) = \dfrac{ab}{\gcd(a,b)}$ for $a,b \in R.$ Does this proof work? Any criticism appreciated: We rewrite $\operatorname{lcm}(a,b) = ...
0
votes
0answers
7 views

Prove of disprove: If f is twice differentiable on (-1,1) and f"(0)>0, then there is δ>0 such that f is convex on (- δ, δ) [duplicate]

First of all I feel like this statement is true. The way I'm thinking about it is that f"'(x)>0 implies convexity. The fact that they used f"(0)>0 makes no difference I think. Even if we consider it, ...
1
vote
1answer
14 views

Discrete Structures: Trying Correcting my Predicate Logic with the appropriate quantifiers

I am trying to correctly use predicate symbols and using the appropriate quantifiers were I have to write each English language statement in predicate logic and the domain is the whole word. $P(x)$ ...
1
vote
2answers
20 views

Why is $\frac{d^m}{d(-x)^m}=(-1)^m \frac{d^m}{dx^m}$

Im not a mathematician so dont judge me with my following "proof". I want to show that a Legendre polynomial is odd if its index is odd, and even if its index is even. This in order to prove that ...
2
votes
2answers
36 views

Prove that Set B is countable - Is this proof correct?

It seems that I have some issues with the rigor of this proof and I don't know what I'm doing wrong. Could someone tell me if this proof is correct and rigorous enough? Here's the question Prove ...
4
votes
0answers
63 views

Linear Algebra Proof conformation

I am trying to prove this theorem in my book. Because it is provided without proof, please let me know what you think! $\mathbf{Theorem:}$ Let V be a finite , $n$ -dimensional vector space and let U ...
6
votes
2answers
77 views

Let R be a ring. Prove that if $x^2=x$ for each $x \in R$, then R is a commutative ring. [duplicate]

Let R be a ring. Prove that if $x^2=x$ for each $x \in R$, then R is a commutative ring. Ok, so I'm just looking for some confirmation that I'm doing this correctly. If we suppose $x,y \in R$ Let's ...
4
votes
1answer
42 views

prove that $\frac{1}{x}$ is not uniformly continuous on $(0,1)$

I would like to show that the function $\frac{1}{x}$ is not uniformly continuous on $(0,1)$ using two approaches. First Approach: We have the fact that if a function $f$ is uniformly continuous on ...
1
vote
4answers
60 views

Solution verification: Prove by induction that $a_1 = \sqrt{2} , a_{n+1} = \sqrt{2 + a_n} $ is increasing and bounded by $2$

I have the following recursive relation (sequence): \begin{align} a_1 = \sqrt{2}, \quad a_{n+1} = \sqrt{2 + a_n} \end{align} My Try: I'm a little skeptical of my manipulations near the end but it ...
2
votes
1answer
34 views

Use Induction to Show $(1+a)^n \ge 1 + na$

If $a$ $\in$ $\mathbb R$ $\ni$ $a > -1$, then ($\forall n$ $\in$ $\mathbb R$) ($(1+a)^n \ge 1 + na$) My main concern is twofold: Firstly, I am concerned that constant $a$ in the proposition may ...
1
vote
1answer
24 views

Prove that the sequence $(b_n)$ converges

Prove that if $(a_n)$ converges and $|a_n - nb_n| < 2$ for all $n \in \mathbb N^+$ then $(b_n)$ converges. Is the following proof valid? Proof Since $(a_n)$ converges, $(a_n)$ must be bounded, ...
2
votes
0answers
21 views

Product Spaces: Tube Lemma

The book I am using for my Introduction of Topology course is Principles of Topology by Fred H. Croom. My professor asked to prove the following Lemma. The Tube Lemma: Let $K$ be a compact ...
0
votes
0answers
31 views

Find all non-trivial submodules of a direct sum of two non-isomorphic simple modules

Let $R$ be a ring with $1$. Let $M_1$ and $M_2$ be two non-isomorphic simple (nonzero) $R$-modules. Find all non-trivial submodules of $M_1 \bigoplus M_2$. Solution: $M_1 \bigoplus M_2 \cong M_1 ...
0
votes
0answers
12 views

Verification of argument for global asymptotic stability of equilibrium point

I am working on proving that an equilibrium point of a two-dimensional dynamical system is globally asymptotically stable. The background and justifications are below. I have gotten to the final steps ...
1
vote
0answers
38 views

Hilbert Space is not locally compact.

The book I am using for my Introduction of Topology course is Principles of Topology by Fred H. Croom. Show that Hilbert Space is not locally compact at any point. This is what I understand: ...
2
votes
0answers
40 views

Proving that $\lim\limits_{x \to -1}(3x^2-3)\sin(x) = 0$.

Prove that $\lim\limits_{x \to -1}(3x^2-3)\sin(x) = 0$. So, by the definition i have to prove that $$ \exists\delta>0 \text{ such that} $$ $$ 0<|x+1|<\delta \longrightarrow ...
0
votes
0answers
38 views

Chebyshev representation of polynomial

In Carl de Boor's A Practical Guide to Splines (1978) problem II.3.a demands a proof that a polynomial $P_ng$ of order $n$ which agrees with a function $g:\mathbb{R}\rightarrow\mathbb{R}$ at $\tau_1, ...
0
votes
1answer
36 views

Proving $2n-8<n^2-8n+14$ for all $n\geq 7$ by induction

For what values of the natural number $n$ is $2n-8 < n^2-8n+14$? (must use induction) I have determined that $n$ appears to work for all values except $n=4,5,6$. I was wondering if this proof ...
2
votes
1answer
17 views

Weak and vague convergence of normal distribution

Let $\mu_n = \mathcal{N}(0,n)$ be the normal distribution with mean $0$ and variance $n$ on $\mathbb{R}$, $\nu$ the zero-measure (which is defined by $\nu(A) = 0$ for any ...
1
vote
2answers
206 views

Can a non-commutative ring contains identity?

Can a non-commutative ring $R$ contains identity? Suppose $R$ contains the identity element 1. Construct an ideal $Z(R) = \{a \in R \mid ra = ar\text{ for all }r \in R\}$. Since $1 \in Z(R)$, $R = ...
4
votes
4answers
89 views

Prove $\sum_{i=2}^{n}\frac{1}{(n-1)n}$ = $\frac{(n-1)}{n}$ using induction.

I need to prove $\sum_{i=2}^{n}\frac{1}{(i-1)i}$ = $\frac{(n-1)}{n}$ using induction. I am getting stuck midway through the inductive step. Here is what I have: $\forall n\geq 2$, where ...
0
votes
1answer
39 views

I am trying to use proof of sequence correctly to make valid

Here I am trying to use a proof sequence so that the argument is valid (hint: the last A’ has to be inferred). (A → C) ∧ (C → B') ∧ B → A' Here are my steps I tried but not sure if this is correct ...
1
vote
2answers
25 views

If $R$ is a ring with unit element $1$ and $\varphi$ is a homomorphism of $R$ onto $R'$, prove that $\varphi(1)$ is the unit element of $R'$.

Herstein 3.4.20: If $R$ is a ring with unit element $1$ and $\varphi$ is a homomorphism of $R$ onto $R'$, prove that $\varphi(1)$ is the unit element of $R'$. I don't understand why $\varphi$ needs ...
0
votes
0answers
42 views

Trying to justify each step correctly in proof sequence

I am trying Justify each step in the proof sequence below for correctly with [A → (B ∨ C)] ∧ B' ∧ C' → A' So I justified my steps here but I am not sure at 1 to 3 if I did it correctly. A → (B ∨ ...
0
votes
1answer
35 views

Prove $a^2+6a+1\perp 375$ for all $a\in \mathbb{Z}$.

Prove $A=a^2+6a+1\perp 375$ for all $a\in \mathbb{Z}$ I thought to write $375=3\cdot5^2$. So if $A$ is coprime with $3\cdot5^2$ they must share no prime factors. Then I test if $3$ or $5$ divide $A$ ...
1
vote
1answer
9 views

Let $R$ be a commutative ring with 1. If $R$ is a PID, then every prime ideal is either zero or maximal.

Let $R$ be a commutative ring with 1. If $R$ is a PID, then every prime ideal is either zero or maximal. My proof: Let $I = (p)$ be a non-zero prime ideal of $R$. Note that $p$ is prime. Since $R$ is ...
1
vote
1answer
26 views

Can independence of a system and a vector be establish if there is only cross-indepedence?

Say that I have the following linear system: $$[A a'] \begin{bmatrix} x \\ x' \\ \end{bmatrix} =Ax + a'x' $$ I want to know when this system is zero if and only if ...
0
votes
1answer
29 views

Function on a well-ordered set

Let $(W,<)$ be a well ordered set. Let $f : W\rightarrow W$ be a function such that $u < v$ implies $f(u) < f(v)$. Show that $\forall w \in W, w \leq f(w)$. I was thinking to consider $T=\{x ...
0
votes
0answers
14 views

Prove by induction: $E[\sum_{i=1}^nc_iU_i(X)]=\sum_{i=1}^nc_iE[U_i(X)]$ Please just check what I've done

Prove by induction: $$E[\sum_{i=1}^nc_iU_i(X)]=\sum_{i=1}^nc_iE[U_i(X)]$$ Let me show you what I've done. I think I'm right: $$n=1,$$ $$E[c_1U_1(X)] = c_1E[U_1(X)]$$ Okay so maybe this one looks ...
0
votes
1answer
32 views

Provide an example to show that $S$ may not necessarily be a unique factorisation domain when $R$ is a unique factorisation domain.

Let $R$ and $S$ be integral domains, and suppose that $\phi:R \rightarrow S$ is a surjective ring homomorphism. Provide an example to show that $S$ may not necessarily be a unique factorisation domain ...
3
votes
2answers
46 views

Proof that an infinite subset of $\mathbb{N}$ is countable

I want to prove that if $A$ is an infinite subset of the natural numbers, then it is countable. I thought of an informal proof: put the elements of $A$ in increasing order. Then associate the ...
2
votes
1answer
26 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 ...
0
votes
1answer
32 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 ...
1
vote
1answer
29 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} ...
5
votes
3answers
51 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 ...
0
votes
0answers
28 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 ...
2
votes
0answers
53 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
31 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
27 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
116 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
31 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
42 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
33 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
31 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
20 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 $N = 2Z_6 = \{2,4\}$. ...