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Questions tagged [proof-verification]

For questions concerning a specific proof or a specific solution, asking for verification, identifying errors, suggestions for improvement, etc. (You should not use this tag if the question does not contain your proposed proof/solution.)

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8 views

Measure of $C^1$ path in $\mathbb{R}^2$

I started studying multivariable integration and still trying to grasp the conecpt of the measure. I`m doing excersices and I keep getting the feeling im doing something wrong so I hope one of you ...
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0answers
19 views

Show third differentiation at point 0 less than 2

Let $f, f', f''$ be continuous real functions on $R$. Suppose that $|f(x)|\leq x^2$ for all $x$ in a neighborhood of 0. To show $|f''(0)|\leq 2$. I used $f''(x)=\lim_{h\to0}\frac{f(x+h)+f(x-h)-2f(x)}{...
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1answer
8 views

Let M be a matching of T, prove that M< N(T) - $\triangle$ (T).

This question is part C of another question: (a) Using induction on $n$, prove that $T$ has at least $\Delta(T)$ leaves. (b) Prove that $B'(T) \geq \Delta (T)$. (c) Let $M$ be a matching of $T$, ...
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0answers
14 views

let the girth of G is at least 2k. Prove that the diameter of Gis at least k.

The question is to prove that if graph G has at least one cycle and that the girth of G (Girth= length of shortest cycle) is at least 2k, then the diameter of G is at least k. My attempt: I tried ...
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0answers
9 views

$f(x,y)=(g(x,y),h(x,y))$ is continuous at $P_0$ iff both $g,h$ are continuos at $P_0$.

Let $f:\mathbb R^2\to \mathbb R^2$, $f(x,y)=(g(x,y),h(x,y))$ where $g,h :\mathbb R^2 \to \mathbb R$. $f$ is continuous at $P_0$ iff both $g,h$ are continuos at $P_0$ My attempt: $\Rightarrow$: ...
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2answers
16 views

Proof by induction coins

Question: Tom only have 2 type of coins: coins: 4 cents and 5 cents. Write a proof by induction that every amount n ≥ a can indeed be paid with Tom coins 1) Base Case: Tom can pay $12, $13, $14, $15, ...
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2answers
17 views

Normed linear space..

In Walter Rudin's Complex Analysis ,it states that by definition$$\|\Lambda\|=\text{sup}\{\|\Lambda x\|: x\in X, \|x\|\leq1\}$$ If $\|\Lambda x\|<\frac{\epsilon}{\delta}\|x\|$ How to show that $\|...
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3answers
13 views

Induction proof verification?

Using induction, I have to prove that $n^2+n$ is divisible by 2. Here's how I did it, and I wanted to know if this is considered valid. I started with $(k+1)^2+(k+1)$, and after simplifying and ...
2
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2answers
42 views

Axioms of algebraic structure - ring

If in the definition of ring $(R,+,\times$) we insist that it has unit element $1$. Then we can show that addition $(+)$ is commutative operation. However, most of the proof which I've seen in MSE use ...
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1answer
19 views

Showing that an injective Darboux function is strictly monotone.

I was hoping someone could tell me how to prove the following problem I was given: Let $f:[a,b]\to\mathbb{R}$ be a function such that for every $y\in[f(a),f(b)]$ there exists $x\in[a,b]$ such that $y=...
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1answer
35 views

Verification that $\lim_{x \to a}\frac{f(x)-f(a)}{x-a}$ and $\lim_{h\to 0}\frac{f(a+h) -f(a)}{h}$ are equivalent definitions of the derivative.

I wanted to verify that for definition of the derivative it is true that: $$ \lim_{x \to a}\frac{f(x)-f(a)}{x-a}= \lim_{h\to 0}\frac{f(a+h) -f(a)}{h}$$ If I denote $h=x-a$, we can let $x\to a$, ...
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0answers
16 views

Need help answering exam questions about Search trees [on hold]

I just got back the exam paper but without the answers, I want to compare what I did in the paper to what you guys would say the answer would be. This is one of the questions of the final year of ...
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1answer
20 views

Given $\lim_{n\to\infty}x_n = \infty$ show that $y_n = \left\{ \sum_{k=1}^n x_k\right\}$ is an unbounded sequence.

Given: $$ \lim_{n\to\infty}x_n = \infty $$ show that $$ y_n = \left\{ \sum_{k=1}^n x_k\right\} $$ is an unbounded sequence. Intuitively this is obvious, however I'm having a hard time ...
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1answer
18 views

Prove that the even cycles are 2-list-colorable.

I need to prove the statement in the title. It's very easy to show that with a particular 2-list-assignment, we can have a proper 2-list-coloring (e.g. with a greedy coloring algorithm). I'm just not ...
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0answers
16 views

Prove that the solution of $x'(t)=Ax+\frac{\sin t}{t+1} (1,\cdots,1)^T$ is bounded

I am trying to show that the solution $x(t)$ of $$x'(t)=Ax+\frac{\sin t}{t+1} (1,\cdots,1)^T$$ is bounded on the interval $(0,\infty)$, where the eigenvalues of the $n \times n$ matrix $A$ have ...
1
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0answers
26 views

Using least common multiple to prove there exists a prime between $2x$ and $3x$

Let $\text{lcm}(x)$ be the least common multiple of $\{1,2,3,\dots, x\}$. Hanson showed that $\text{lcm}(x) < 3^x$ I'm wondering if the following argument is valid for showing that there is ...
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1answer
28 views

Proof for sum of digits of a number until sum is a single number

Here is a more elaborate description of the problem statement. What I found with a few examples is that given a number, say 569. If we are required to sum its digits repetitively until the sum is ...
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1answer
32 views

Check if $\sup\left\{\int_J|f(x)|\chi_B(x)\, dx:\lambda(B)\le1/k\right\}\to 0$

Let $J\subset\Bbb R$ be a perfect interval and let $f\in\mathcal L_1(J,E)$. Set $$R_k:=\sup\left\{\int_J|f(x)|\chi_B(x)\, dx:\lambda(B)\le1/k, B\subset J\right\}\tag1$$ I want to see if it is true ...
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1answer
39 views

A counterexample of Banach Steinhaus Theorem

I was reading about a consequence of Banach-Steinhaus theorem which states that: Let $E$ be a Banach space and $F$ be a normed space, and let $\{T_n\}_{n\in \mathbb{N}}$ be a sequence of bounded ...
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1answer
23 views

If $\aleph_\alpha=\alpha$, then $\alpha$ is a limit ordinal

If $\aleph_\alpha=\alpha$, then $\alpha$ is a limit ordinal. My attempt: Assume the contrary that $\alpha$ is not a limit ordinal. Then $\alpha$ is a successor ordinal and thus $\alpha=\beta+1$ for ...
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0answers
37 views

No group of order 10000 is simple

A proof of this fact was already given here: No group of order 10,000 is simple However, I am wondering whether or not the following proof works as well: By way of contradiction, suppose $G$ is ...
1
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2answers
34 views

Show that $ \inf_{c \in \mathbb{R}} \|f - c\|_{\infty} = \| f - \frac{1}{2}(\min f + \max f) \|_{\infty}$

I'm very unsure about what I did here, so if someone could proof check on the following, I would be very thankful. Let $I$ be a an interval of $\mathbb{R}$. For $f \in \mathcal{C}(I)$, we say that $...
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0answers
13 views

If a probability measure has two densities, then they agree a.e. (Proof verification)

Suppose that $f, g$ are two nonnegative measurable functions that integrate to one over $\mathbf{R}$, with respect to Lebesgue measure and are such that the probability measure $\mu(A) = \int_A f d\...
3
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1answer
35 views

Showing that monotone functions have at most countable discontinuities.

I want to show that a map $F: \mathbf{R} \to \mathbf{R}$ has at most countable discountinuities, if $F(x) \leq F(y)$ whenever $x \leq y$. Here's the idea. Let's use standard notation $F(x^+), F(x^-)$...
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6answers
140 views

Proof verification for $\lim_{n\to\infty}\frac{1}{n}(1+\sqrt2+\dots + \sqrt{n}) = +\infty$

Show that: $$ \lim_{n\to\infty}\frac{1}{n}(1+\sqrt2+\dots + \sqrt{n}) = +\infty $$ I've tried the following way. Consider the following sum: $$ \sqrt n + \sqrt{n-1} + \dots + \sqrt{n-\frac{n}{2}} +...
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0answers
36 views

Did I apply the Ceva's theorem correctly to this problem?

I need to confirm the following solution. I'm making a mistake somewhere. But I can't find the error. I apply the trigonometric form of the Ceva's theorem: $$\frac{\sin \angle 3}{\sin \angle 4}× \...
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0answers
14 views

Associative property in a ring

Let $R$ a ring and we define $$\text{seq}R=\{f=(a_0,a_1,\dots,)\;|\;a_i\in R\}.$$ On this set we define the following operation: let $f=(a_n)_{n\ge1}$ and $g=(b_n)_{n\ge1}$ $$fg=(c_0,c_1,\dots,),$$ ...
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1answer
127 views

A proof that $\sqrt{2}$ is not a rational number.

Is this proof correct? Suppose that $\sqrt{2}=\frac{a}{b}$, where $a,b \in \mathbb{N}$ and $a$ is as small as possible. Then $\sqrt{2}b=a$ which means $2b=\sqrt{2} a$. So we rewrite $\sqrt{2}=\frac{a}...
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2answers
44 views

Why is my proof that $\mathbb R$ is disconnected wrong?

The definition of connectedness in my notes is: A topological space $X$ is connected if there does not exist a pair of non empty subsets $U$, $V$ such that $U\cap V=\emptyset$ and $U\cup V=X$. ...
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1answer
13 views

Is $ \max_{x\in\mathbb{R}^n} \{ f(x)+g(x) \} = \max_{x\in\mathbb{R}^n} f(x)+\max_{x\in\mathbb{R}^n} g(x) $ if $f$ and $g$ are affine in $\mathbb{R}$?

Let $x \in \mathbb{R}^n$, and let $f(x)$ and $g(x)$ be two affine functions in $\mathbb{R}$. Is the following property true? $$ \max_{x\in\mathbb{R}^n} \{ f(x) + g(x) \} = \max_{x\in\mathbb{R}^n} f(x)...
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2answers
42 views

$1 < a$ and $b\ne0$ imply $1<a^b$

$1 < a$ and $b\ne0$ imply $1<a^b$ when $a,b$ are arbitrary nonnegative integers. I've tried to prove it by induction. I've assumed that $b < a$ (Is valid my assumption?) I'm using this ...
1
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1answer
22 views

A closed (?) subset of the set of probability measures. Is my reasoning correct?

Let $f:\mathbb{R}^{d} \rightarrow [0,\infty) $ continuous function and denote by $P$ the space of probability measures on $\mathbb{R}^{d}$ and by $P_R \,=\, \{\nu \in P \, | \, \int f d \nu \leqslant ...
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4answers
66 views

Prove that if $A^2=0$, then $0$ is the only eigenvalue of $A$. [duplicate]

Does my proof hold up to prove that $0$ is the only eigenvalue of $A$ if $A^2 = 0$? Let $A$ be an $n \times n$ matrix. $A^2 = A*A$ because of matrix multiplication. If $A = k$, where $k \neq 0$, ...
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2answers
39 views

$a = d$ implies $a^b = d^b$

Prove that $a = d$ implies $a^b = d^b$, where $a, d$ are arbitrary nonnegative integers and $b$ is any positive integer. If I could use division I think it could be something like that: $a^b / d^b ...
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1answer
11 views

Prove derivative of order $n$ at neighbourhood $x_0$

If $$f(x)-f(x_0)=g(x)(x-x_0)$$ and $g \in C^{(n-1)}(U(x_0))$, where $U(x_0)$ is a neighbourhood of $x_0$, then show that $f(x)$ has a derivative $f^{(n)}(x_0)$ of order $n$ at $x_0$. I think ...
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1answer
77 views

$a < b$ and $c<d$ imply $a+c < b+d$

$a < b$ and $c<d$ imply $a+c < b+d$ when $a,b,c,d$ are arbitrary nonnegative integers. I know that (assuming we include zero) $$\begin{align*} a<b \Leftrightarrow (\exists x\in \mathbb ...
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3answers
24 views

Union of closures

Let $X$ be a topological space $\mathscr{ B}$ be a collection of subsets of $X$. Show that $\overline{ \bigcup \limits_{\alpha \in \mathscr{B}} B_\alpha} \subset \bigcup \limits_{\alpha \in \mathscr{B}...
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0answers
14 views

Am I right in this proof of a criterion for the nonsingularity of a conic curve?

$\newcommand{\C}{\mathcal{C}}$ This is an exercise in Silverman and Tate's Rational Points on Elliptic Curves: Let $\C$ be the conic given by the equation $$ F(x,y)=ax^2+bxy+cy^2+dx+ey+f=0.$$ ...
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1answer
31 views

If $(A,\preccurlyeq)$ is a linear ordering such that $|\{y\in A\mid y\preccurlyeq x\}| < \aleph_\gamma$ for all $x\in A$, then $|A|\le\aleph_\gamma$

If $(A,\preccurlyeq)$ is a linear ordering such that $|\{y\in A\mid y\preccurlyeq x\}| \le \aleph_\gamma$ for all $x\in A$, then $|A|\le\aleph_\gamma$. Does my attempt look fine or contain logical ...
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0answers
30 views

Zorn's Lemma implies Axiom of Choice

Zorn's Lemma implies Axiom of Choice Does my attempt look fine or contain logical flaws/gaps? Any suggestion is greatly appreciated. Thank you for your help! My attempt: Let $S$ be a collection of ...
0
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0answers
20 views

Well-Ordering Principle implies Zorn's Lemma

Well-Ordering Principle implies Zorn's Lemma Does my attempt look fine or contain logical flaws/gaps? Any suggestion is greatly appreciated. Thank you for your help! My attempt: Let $(A,\...
5
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1answer
23 views

Let $a<b<c$, where $a$ is a positive integer and $b$ and $c$ are odd primes. Prove that if $a \mid (3b+2c)$ and $a \mid (2b+3c)$, then $a=1$ or $5$.

The prove I tried is the following. I really wish someone can check if I made some logical mistake, especially the last part I found myself diffident proving $a$ can only be $1$ or $5$. Because $b\...
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0answers
42 views

Solution verification: evaluate $\lim\limits_{n \to \infty}\frac{1^{\lambda n}+2^{\lambda n}+\cdots+n^{\lambda n}}{n^{\lambda n}}$ where $\lambda>0.$

Problem Evaluate $$\lim_{n \to \infty}\frac{1^{\lambda n}+2^{\lambda n}+\cdots+n^{\lambda n}}{n^{\lambda n}}$$ where $\lambda>0.$ Solution Denote $$S_n:=\sum_{k=1}^{n}\left(\frac{k}{n}\right)^{\...
2
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1answer
53 views

How to choose between two solutions where both seem to be correct?

Question to solve: $ ysin(2x)dx - (y^2 + cos^2x)dy = 0 $ -----(i) Writing (i) in exact form [$ M(x,y)dx + N(x,y)dy = 0$], we get $ysin(2x)dx + [-(y^2 + cos^2x)]dy = 0$ $where,M=ysin(2x)$ and $N=[...
1
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0answers
30 views

Axiom of Choice implies Well-Ordering Principle

Axiom of Choice implies Well-Ordering Principle. My textbook only presents the construction of function $F$ and does not provide details on how to define such well-ordering. I try to fill in the ...
0
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1answer
15 views

Determining whether or not a relation involving absolute value is transitive

I needed help doing a relation problem the specific problem is $|x+y|$ = $|x|$ + $|y|$ I first tried thinking of some counter-examples but I couldn't think of any so i tried showing it. So to show ...
2
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2answers
52 views

What are the Legendre symbols $\left(\frac{10}{31}\right)$ and $\left(\frac{-15}{43}\right)$?

I have the following two Legendre symbols that need calculated: $\left(\frac{10}{31}\right)$ $=$ $-\left(\frac{31}{10}\right)$ $=$ $-\left(\frac{1}{10}\right)$ $=$ $-(-1)$ $=$ $-1$ $\left(\frac{-15}{...
0
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1answer
20 views

Prove that $f \in R(F_s)$ on the interval $[a, b]$ and that $\int f dF_s = f(s)$

Let $a<s<b$ and let $f:[a, b] \to \mathbb{R}$ be a bounded function that is continous at the point s. Define $F_s(x) = \begin{cases} 0, & \text{if $a \le x \lt s$} \\ 1, & \text{if $s \...
1
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1answer
25 views

The lobster eating problem, a circular table and the possible table arrangement

The following problem was a fun combinatorics question I encountered, there seems to be a buildup in the question, I wanted to ask if my reasoning sounds plausible. I'm sorry if my reasoning is a bit ...
0
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1answer
22 views

Prove that any divisor of order 0 on non-singular projective curve of genus $g$ is equivalent to other

Could you please check whether the solution below is ok? There is an exercise from Shafarevich's Basic Algebraic Geometry, vol. 1, ex. 7.7.21. Let $o$ be a point of an smooth algebraic curve $X$ of ...