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

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22 views

Prove that $\int (\delta x)=\delta^{-d} \int f$

Let $f$ be a real-valued integrable function on $\mathbb{R}^d$. Prove that $$\int f(\delta x) = \delta^{-d} \int f.$$ I let $f(x)=\chi_E(x)=\begin{cases} 1 & \text{if }\delta x \in E \\ 0 ...
0
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0answers
14 views

Finding the range of a cannonball- proof verification.

I asked such a question before but I do learn best by mistakes and corrections.(I didn't fully understand it yet.) I could really use your verification: A cannonball is being fired with a velocity of ...
0
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0answers
37 views

proof that $\frac{e^{-t}}{2}(t^2+2t+2)\le1$ for $t\ge0$

Show that $\forall t\ge0,x\le1$ where $$x=\frac{e^{-t}}{2}(t^2+2t+2),t\in\mathbb{R}.$$ My proof we have $x=\frac{(t^2+2t+2)e^{-t}}{2}$ then ...
2
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1answer
28 views

Closed points are dense in $\operatorname{Spec} A$

From 3.6.J in Vakil: Let $k$ be a field, and let $A$ be a finitely generated $k$-algebra. We want to show the closed points are dense in $\operatorname{Spec} A$. This is the set of prime ideals of ...
1
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1answer
15 views

Is this simple proof that the Frobenius endomorphism of an elliptic curve defined over $\mathbb F_q$ is surjective valid?

I am quite sure that the following "proof" is flawed, but I don't see why: Let $E$ be an elliptic curve defined over $\mathbb F_q$. Since $E$'s ideal is generated by a polynomial in $\mathbb ...
2
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0answers
33 views

Proving Finiteness

For any set $X$, if $\cup X$ is finite, then $P(X)$ (power set of X) is finite. For any Transitive set $X$, if $P(X)$ is finite, then $\cup X$ is finite. I'm a little confused with these because ...
3
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1answer
40 views

Show that $f$ is continuous at exactly one point

Let $f:\mathbb{R}\to\mathbb{R}$ be defined by $$f(x)= \begin{cases} 5x+7 & \text{ if } x \text{ is rational } \\ x+11 & \text{ if } x \text{ is irrational } \end{cases}$$ ...
1
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1answer
34 views

Use Least Upper Bound to show that $\mathbb{R}$ is completee

Use Least Upper Bound to show that $\mathbb{R}$ is complete. The following is the proof I did, and it's slightly different from what I see from the book. Can someone check if I'm missing ...
2
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1answer
42 views

Prove that $\lim_{x \to a} \big[ f(x)+g(x)\big] = p+q$

Let $f: A \to \mathbb{R}^m$ and $g: B \to \mathbb{R}^m$ be functions such that $A,B \subseteq \mathbb{R}^p$, $a \in \mathbb{R}^p$ and let $\beta$ be a fixed number. Furthermore let $f(x) = ...
0
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1answer
23 views

Can I have a critique of this set theory proof/Advice on a similar proof?

This is an exercise from Mendelson's Introduction to Topology. The first part is to prove, given a function $\ f:A \rightarrow B$, that $\ X \subset f^{-1}(f(X))$ for all $\ X \subset A$. Here's my ...
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1answer
13 views

Prove: Monotonic And Bounded Sequence- Converges

Let $a_n$ be a monotonic and bounded sequence, WLOG let assume it is monotonic increasing. $a_n$ is bounded therefore there is a Supremum, $Sup(a_n)=a$, therefore $a_n<a+\epsilon$. On the other ...
0
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1answer
25 views

Compute the wedge product n times

Let $\omega$ be a 2-differential form in $\mathbb{R}^{2n}$ given by $$\displaystyle \omega=dx^1\wedge dx^2+dx^3\wedge dx^4 + \cdots + dx^{2n-1}\wedge dx^{2n}$$ Compute: $$\displaystyle ...
6
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4answers
79 views

Prove that if $p$ is a prime such that $p^2+2$ is a prime then $p=3$.

My problem in my solution is that I don't know if the operations I apply on congruence modulo n are admissible. I could really use some guiding: Attempt: Let there be $p\ne 3$ fulfilling the ...
0
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1answer
56 views

Verification of solution of a contest problem with a limit of nested radicals

They gave me 0 points for this problem. I think it's unfair. What do you think of this proof, is it correct? $\lim\limits_{n\to\infty} \underset{2n\text{ roots ...
2
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1answer
52 views

Using Fermats prime numbers to prove that there is infinitely many prime numbers

A Fermat number $F_n$ is of the form $F_n = 2^{2^n} + 1$ Furthermore, $F_n = 2 + F_0F_1F_2......F_{n-1}$ Now I already proved that if $n \neq m$ then $\gcd(F_n,F_m) = 1$ Here is the proof Without ...
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3answers
1k views

Prove that there is no smallest positive real number

I have to prove the following: $$\text{Prove that there is no smallest positive real number}$$ Argument by contradiction Suppose there is a smallest positive real number. Let $x$ be the smallest ...
2
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2answers
28 views

Show that $\overline \varphi (a Z (D_4)) = Id$

Consider $$\begin{align}\overline \varphi : \frac{D_4}{Z(D_4)} &\to \frac{D_4}{Z(D_4)} \\aZ(D_4) &\mapsto xax^{-1}Z(D_4)\end{align}$$ where $$D_4 = \{id, \alpha, ...
2
votes
1answer
16 views

Proving $\alpha\colon S\to T$ is one-to-one if $\alpha(A\cap B)=\alpha(A)\cap\alpha(B)$, where $A,B\subseteq S$

Prove that $\alpha\colon S\to T$ is one-to-one if $\alpha(A\cap B)=\alpha(A)\cap\alpha(B)$. Book solution: Assume that $\alpha(A\cap B)=\alpha(A)\cap\alpha(B)$ for every pair of subsets $A$ and ...
1
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2answers
63 views

Is my proof of a seemingly trivial question involving infinite sums correct?

Given that $\sum_{n=0}^{\infty}a_n^2<\infty$ and $\sum_{n=0}^{\infty}b_n^2<\infty$, where $a_n$ and $b_n$ are sequences of real numbers, I'm trying to show that ...
1
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1answer
45 views

Probability of same birthday

I think I solved this problem but I would like to know if I am right or wrong, I am not quite sure. We assume that the year has 365 days and the birthdays are uniformly distributed. We want to find ...
2
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1answer
31 views

Proving that if $S$ has an infinite subset then $S$ is infinite

Definition$\quad$ A set $S$ can be defined as infinite if there exists a mapping from $S$ to $S$ that is one-to-one but not onto. Otherwise, $S$ is finite. Problem: Using the definition of ...
2
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2answers
18 views

Proving that a positive derivative means the function is smaller “to the left” and larger “to the right” for certain values

I was trying to prove that if $g$ is differentiable on an open interval $I$ with $a\in I$ and $g'(a)>0$ then we can find $x<a$ for which $g(x)<g(a)$ and $y>a$ for which $g(y)>g(a)$, I ...
6
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1answer
36 views

Is $f(x)$ reducible if $f(a)=0$

I am confused about this seemingly trivial question: If $f(a) = 0$ for some $a\in D$, then when is $f(x)$ reducible in $D[x]$? ($D$ is an integral domain). My answer: Always. Let $f(a)=0$. ...
0
votes
1answer
29 views

$n$ divides $a_1 - a_2$ as well as $b_1 - b_2$. Show that $n$ divides $a_1b_1 - a_2b_2$.

I keep arriving at $a_1b_1$ and $a_2b_2$ having the same sign if I multiply the equations $a_1 - a_2 = nk$ and $b_1 - b_2= np$ times each other. They must be opposite signs so that I can say that $n$ ...
1
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2answers
28 views

Which of the properties, Reflexive, Irreflexive, Symmetric, Asymmetric, Antisymmetric, Transitive, Linear, does F satisfy?

Let $S={(n,m) ∶n,m∈Z^+}$. Define the relation F on S by ${(n,m),(i,j)}∈F$ if and only if $nj=mi$. In other words, let $F = {((n, m), (i, j)) ∈ S × S: nj = mi}$. Proof F is reflexive: Show that for ...
1
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1answer
22 views

displacement of water

I just read the following puzzle on puzzling SE and it made me wonder, is there a proof which can be formulated to see if an object of any size and weight would cause more displacement inside the boat ...
1
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0answers
30 views

Two statements about partial limits and intervals

Let $(a_n)$ be a sequence and $I=(a,b)$ an open interval such that every $L\in(a,b)$ is a partial limit of $(a_n)$. Decide whether or not the following statements are true: $\{a_n | n \in ...
1
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0answers
13 views

Verify combination of disjoint subsets $C$ and $D$ is onto

Let $C$ and $D$ be disjoint subsets of set $A$ and $f:C→B$ and $g:D→B$. Define a function $h(x)$ as follows: $$ h(x)=\left\{ \begin{array}{c} f(x) \textrm{ if } x∈C \\ g(x) \textrm{ if } x∈D ...
2
votes
1answer
18 views

Finding upper bounds of a set

Suppose $A = \{ x \in \mathbb{R} : x + \frac{1}{x} < 5 \} $. I want to find 2 upper bounds of $A$. My intuition tells me that $5$ is indeed an upper bound. To see this, I want to show that for ...
3
votes
1answer
31 views

Are there general guidelines to make “assumptions” when proving limits?

I am studying the definition of the limit using Paul's Online Notes When proving the following limit (Example 3) $\lim\limits_{x \to 2} x^2+x-11 = 9$ At one point he assumes: $|x+5| < K$ He ...
1
vote
2answers
32 views

How many words with letters from the word ABRACADABRA if they must end in a consonant and $d$ must be after $r$.

How many words with letters from the word ABRACADABRA if they must end in a consonant and $d$ must be after $r$. What I did: I have $A:5$ $B:2$ $R:2$ $C:1$ $D:1$ If the words must end in a ...
3
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1answer
20 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 ...
1
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2answers
23 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
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1answer
69 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
41 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 ...
1
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0answers
31 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 ...
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0answers
31 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) = ...
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0answers
8 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
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1answer
20 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
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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
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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
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1answer
77 views

Linear Algebra Proof confirmation

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
1answer
84 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
43 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
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4answers
64 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
23 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
40 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 ...
1
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0answers
39 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: ...