For questions about the formulation of a proof, not about the mathematics behind it.

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3
votes
2answers
69 views

Existence of a normal subgroup in G

Today on my algebra test I had such an exercise: Let $|G|=66$. Show that there is a normal subgroup in $G$ of order $3$. I am not even sure that's true. I wanted to show that $n_{3}$=1. But from ...
0
votes
2answers
71 views

Prove that a set infinite if it has infinite proper subset

Suppose that $A$ is an infinite set and $A \subsetneq C$. Use the definition of "infinite set" to prove that $C$ is infinite also. I am trying to prove that $C$ is infinte. Definition (Infinite ...
0
votes
3answers
184 views

Prove the derivative of $\sin(1/x)$ exists

How do I prove the derivative of $$\sin(1/x)=-\frac{1}{x^2}\cos(1/x)$$? I understand that you use $$f'(x_0) = \lim_{x \to x_0} \frac{\sin(1/x) - \sin(1/x_0)}{x-x_0} = -\frac{1}{x_0^2}\cos(1/x_0)$$ ...
1
vote
4answers
273 views

Prove that the Fibonacci recursion diverges

I have this sequence with $ n \in \mathbb{N} $ $ f(1) = f(2) = 1 $ and $ f(n) = f(n-1) + f(n-2) $ for $n \ge 3$ I think this sequence is bounded below and unbounded above. So it's clear that this ...
8
votes
1answer
226 views

question on translation of operator proof

Has anyone studied the book 'Nonlinear Partial differential equations with applications' by Tomas Roubicek? I am interested in discussing a point of interest in this book. Specifically, on page 52, ...
1
vote
2answers
45 views

Manipulation of combinations

Let $k,n\in\Bbb N_0$, with $k\le n$. Prove that $$\binom{n+1}{k+1}=\sum_{j=0}^{n-k}\binom{n-j}k\;.$$ Just was hoping someone could give me a hint or two with this problem. I think it has to ...
2
votes
4answers
444 views

Establish the convergence and find the limits of the following sequence

$a_n = \left(1+\dfrac{1}{n}\right)^{n+1}$ I know that the answer is supposed to be $e$ but I am unsure how to reach that answer. I am so lost where to even begin with this
1
vote
0answers
28 views

Prove that $Q_{8}'=\{1,-1\}$. Is my proof correct?

Prove that Prove that $Q_{8}'=\{1,-1\}$. My proof: $Q_{8}'\neq \{1\}$, because $Q_{8}$ is not abelian. $|Q_{8}/\{-1,1\}|=4$. So $Q_{8}/\{-1,1\} \cong \mathbb{Z}_4$ or $Q_{8}/\{-1,1\} \cong ...
1
vote
1answer
35 views

Showing that G is solvable

Let $|G|=200$. Show that G is solvable. My beginning of the proof: $|G|=200=2^3*5^2$ Let $n_5$ be the number of Sylow p-subgroups. Then $n_5|8$ and $n_5\equiv1 mod5$. And it implies that $n_5=1$. ...
3
votes
1answer
29 views

Proof: Characterize m

Characterize $m$, an integer, such that $m^2≡1 \pmod{5}$. State your characterization as an "if and only if" statement and then prove it. This question is on my study guide for a test that is on ...
1
vote
1answer
38 views

Showing that $A \subseteq B$ for $A=\{6t\mid t \in \mathrm Z\}$ and $B=\{3t\mid t \in \mathrm Z\}$

Let $A=\{6t\mid t \in \mathrm Z\}$, and $B=\{3t\mid t \in \mathrm Z\}$. Then, show $A$ is a subset of $B$ and prove or disprove that $A = B$. I already know that $A \neq B$, for I can pick a ...
0
votes
1answer
126 views

Prove: The relation $R$ on $\mathbb{N}$ is reflexive, symmetric and transitive

Prove: The relation $R$ on $\mathbb{N}$ given by $mRn$ iff there are natural numbers $p$, $q$ with $m^p$ = $n^q$ is reflexive, symmetric and transitive. Proving $R$ is reflexive: Proof. Suppose $m$ ...
3
votes
2answers
107 views

Prove every maximal ideal of $A$ is a prime ideal (Hint: Use the fact that $J$ is a maximal ideal iff $A/J$ is a field.)

Let $A$ be a commutative ring with unity. Prove: Proof every maximal ideal of $A$ is a prime ideal (Hint: Use the fact that $J$ is a maximal ideal iff $A/J$ is a field.) In the question before ...
0
votes
1answer
143 views

Proof by-contradiction that $(A \subseteq B) \implies (A \setminus B = \{ \})$

I'm studying for an exam and I'm having trouble with one of these problems. Use proof-by-contradiction to prove the predicate $$(A \subseteq B) \implies (A \setminus B = \{ \})$$ where A and B ...
1
vote
1answer
82 views

Is this proof about clock hands lining up correct?

Is http://joshuaoldenburg.com/articles/clock-hands-line-up/ a proof? I.e. does it sufficiently prove the times where the clock hands line up? $$ \begin{align} H &= \text{hour (1-12)} \\ M &= ...
2
votes
2answers
101 views

Prove $1 + \frac{1}{4} + \frac{1}{9} + \cdots + \frac{1}{n^2} < 2 - \frac{1}{n}$ by Induction [duplicate]

The Question Prove $1 + \frac{1}{4} + \frac{1}{9} + \cdots + \frac{1}{n^2} < 2 - \frac{1}{n}$ where $n\ge2$ and $n$ is an integer by Induction My Work Basis Step: 1 + $\frac{1}{4} = ...
1
vote
3answers
43 views

Show that $n!<n^n $ where $n>1$ and is a Positive Integer

Basis Case: $2! = 2\times1 = 2$ $2^2 = 4>2$ Inductive Hypothesis: $k!<k^k$ Induction Step: $k!<k^k$ $k!(k+1) < k^k(k+1)$ $(k+1)! < k^{k+1} + k^k$ I'm confused on where to go ...
0
votes
1answer
154 views

Proving the formula for the cardinality of cartesian products.

Consider sets A and B where |A|=m and |B|=n. Prove by induction on n for a given m that |AB| = mn for all m,n ∈ N where AB = cartesian product. My attempt : Base Case - consider when n = 0 so B is ...
0
votes
1answer
51 views

Proving Bijectivity and Finding the inverse of a function.

I am given this problem: Suppose a, b, c, d ∈ R and ad − bc ≠ 0. Define f : R\ {$d\over -c$} → R\ {$a \over c$} by f(x) is $ax+b \over cx+d$. How do I prove that it is injective and surjective? ...
3
votes
2answers
88 views

Prove that $\Bbb N × \Bbb N$ is countable.

I am given this problem: Prove that $\Bbb N × \Bbb N$ is countable by using the function $f(m, n) = 2^m3^n$ and Theorem that says any subset of a countable set is countable. I'm not exactly sure how ...
0
votes
1answer
20 views

Complex numbers property proof. [duplicate]

I eas given this quesstion in one of my Linear Algebra course with the excercises regarding minimal polynomialsm eigenvalues and diagonalizable matrix: Show that for any two numbers $a,b \in ...
0
votes
4answers
67 views

$|(0,1)| = |\mathbb R|$

For this problem in proving that the cardinality of (0,1) is equal to that of the set of real numbers, would I just prove that (0,1) is uncountable, and then use the theorem that the subset of an ...
1
vote
2answers
39 views

Proving function is not onto

Let A represent the set of real numbers other than $-1$. Consider $f: A \to R$ defined by $f(a) =\frac{ 2a}{ a + 1}$ How would I prove this function is not onto?
2
votes
3answers
35 views

Confused about limit proofs conceptually

In a question like this: Prove that if $\lim_{x \to a} f(x) = l$ and $\lim_{x \to a} g(x) = m$ then $\lim_{x \to a} \max(f(x), g(x)) = \max(l, m)$ In general, when asked for proofs like this, are ...
1
vote
0answers
54 views

Proof of Correctness: Recursion inside loop

I am trying to prove the correctness of the algorithm in the research paper. It is at page 17 in the pdf. ...
0
votes
1answer
54 views

Prove The Limit Does Not Exist

So I have a few questions in which I have to prove that the given sequence does not have a limit and I'm not too sure if I'm on the right track and if I am what is the next step that I have to do. Can ...
6
votes
5answers
931 views

Prove that there is no number that divides both n and n+1

Statement There is no number $x > 1$ that divides both $n$ and $n+1$. Proof (my attempt) Indirect proof: \begin{align} x\mathbin{\vert} n & \implies n = xt_1 \\ x\mathbin{\vert}(n+1) ...
1
vote
2answers
95 views

Show every bounded infinite set has a maximum limit point and a minimum limit point.

Show every bounded infinite set has a maximum limit point and a minimum limit point. Here is my thought even if it is not formal Let $S$ be bounded and infinite set. Bolzano–Weierstrass theorem: ...
9
votes
3answers
506 views

Instructive examples of elegant, clear, rigorous, terse, but “non-dull” mathematical prose

On the "About" page of the Mathgen project one can read: "More seriously, I think this project says something about the very small and stylized subset of English used in mathematical writing. ...
1
vote
0answers
66 views

Writing mathematical proof

i have just finished Multivariate calculus and so far all the mathematics i have done are those calculations sort of questions. However i began to realize that it is not the proper way to do maths and ...
0
votes
2answers
84 views

Suppose G is a group, p is prime , Then the number of elements of G of order p is multiple of (p-1) [closed]

I need Help . "Suppose $G$ is a group, $p$ is prime , Then the number of elements of G of order $p$ is multiple of $(p-1) $". Give me any advise or note
3
votes
1answer
211 views

Prove that two segments are congruent in the arbelos

Background Info + Problem I teach HS Geometry to middle school age students. I generally like to try to solve problems instead of looking up the answer, but this week a student emailed me a problem ...
-1
votes
2answers
97 views

Big-Oh and limits proof?

Prove or disprove: $2^n$ is in $O(3^n)$. I know I have to use some calculus limit techniques but I can't seem to get anywhere. Steps and an approach would be helpful, especially confirming if this has ...
0
votes
1answer
114 views

Big-Omega proof using L'Hopital's Rule?

Prove or disprove: $15n^2$ is in $\Omega(3 \times 2^n)$ So we'd have to prove or disprove this statement: $$ \exists c \in\mathbb{R}^+,\,\exists B\in\mathbb{N}, \forall n \in\mathbb{N}, n ≥ B ...
1
vote
1answer
22 views

Correctly defined measure

I need to show that the measure is unique (correct definition). Prove that the function $\lambda: \sigma(\mathcal{A}\cap V) \rightarrow [0,1]$ such that $\lambda[(A\cap V)\cup(B\cap V^c)]:=\mu(A)$ ...
0
votes
2answers
53 views

If H and K are finite subgroups of G (another proof )

I have a question and it's solution , but I want another proof if there exist . Thanks
3
votes
2answers
117 views

If $a+1/a$ is an integer, then so is $a^t+1/a^t$ for $t\in\mathbb N$

I need to show if $a$ is in $\mathbb{R}$ but not equal to $0$, and $a+\dfrac{1}{a}$ is integer, $a^t+\dfrac{1}{a^t}$ is also an integer for all $t\in\mathbb N$. Can you provide me some hints please?
2
votes
2answers
81 views

A positive integer $n$ is such that $1-2x+3x^2-4x^3+5x^4-…-2014x^{2013}+nx^{2014}$ has at least one integer solution. Find $n$.

A positive integer $n$ is such that $$1-2x+3x^2-4x^3+5x^4-...-2014x^{2013}+nx^{2014}$$ has at least one integer solution. Find $n$.
0
votes
1answer
189 views

Use Resolution to proove a sentence in First Order Logic

I was just wondering if anyone could tell me if I've solved this problem right. If wrong, I would like to know what I did wrong. "Use resolution to prove Green(Linn) given the information below. You ...
3
votes
0answers
95 views

Real analysis question: Suprema and Infima

Let $S$ be an ordered set with the $L.U.B$ Property, $S \supset B \neq \varnothing$, $B$ is bounded below. Write $L = \{ l : l \; \text{is a lower bound of } \; B \} $. Then, it follows that ...
0
votes
1answer
52 views

Did I do this big-Omega proof correctly?

Prove or disprove: 6n^3 – 4n^2 + 3n +2 is in Ω (5n^3 – n^2 + n +1). So I'm not sure if I did this right or not, any pointers or the correct steps would be helpful Ǝc ∈ ℝ+, ƎB ∈ ℕ, ∀n ∈ ℕ, n ≥ B ⇒ ...
2
votes
2answers
196 views

How to prove/disprove proof on limits (delta-epsilon)

Prove or disprove: $$ \forall \epsilon > 0, \exists \delta>0, \forall x, y \in \mathbb{R}^+, |x - y| > \delta ⇒ |x + y| > \epsilon $$ I've been trying this for some time now but can't ...
1
vote
1answer
61 views

big-Oh prove or disprove 2^n is in big-Oh(3^n)

the definiton of Big-Oh says $\exists c\in$R+,$\exists B\in$ N,$\forall n\in$N, $n \geq B$$\implies$$2^n \leq c\times 3^n$. I believe $2^n \in O(3^n)$, but how to prove it? can anyone help. This this ...
-2
votes
2answers
38 views

Proving a Relation that is a Function by Division Algorithm [duplicate]

Let A=B=$\mathbb{N}$ R is: (a,b)$\in$R iff for some q$\in$Integers a=5q+b WHERE 0$\leq$b<5 Given a relation, show that it's a function. To Show: 1) $\forall$a$\in$A$\exists$b$\in$B((a,b)$\in$R) ...
0
votes
1answer
120 views

Contradiction proof for a limit law $f(x) \le g(x)$

Suppose that $f(x) \le g(x)$ for all $x$. Prove that $\displaystyle \lim_{x \to a} f(x) \le \lim_{x \to a} g(x)$, provided these limits exist. I posted a similar question, but this is a different ...
2
votes
2answers
37 views

Proof by Induction - Wrong common factor

I'm trying to use mathematical induction to prove that $n^3+5n$ is divisible by $6$ for all $n \in \mathbb{Z}^+$. I can only seem to show that it is divisible by $3$, and not by $6$. This is what I ...
5
votes
1answer
106 views

With $N$ a constant $>0$, show $\prod_{p<x}\frac{1}{p^{N+1}-1}>\frac{0.2}{\log^2 x}$.

Related. Show that if $x$ is large enough,$$\prod_{\substack{p<x \\ p \ \text{prime}}}\frac{1}{p^{N+1}-1}>\frac{0.2}{\log^2 x}.$$ Speaking of which, Theorem 6.12, and maybe others, of this ...
0
votes
3answers
146 views

Proof by minimum counter example

I need to prove that $n^4-n^2$ is divisible by 12 by minimum counter example. I understand the process but I don't understand how we arrive at m>=7. I have seen different proofs but I still don't know ...
7
votes
3answers
115 views

valid proof of series $\sum \limits_{v=1}^n v$

$$\sum \limits_{v=1}^n v=\frac{n^2+n}{2}$$ please don't downvote if this proof is stupid, it is my first proof, and i am only in grade 5, so i haven't a teacher for any of this 'big sums' proof: if ...
0
votes
2answers
97 views

GCD proof using fundamental theorem of arithmetic

prove: $\gcd(m,n)=1$ if and only if $\gcd(m^i,n^r)=1$ I believe you need to do something with fundamental theorem of arithmetic to prove one of the sides. Not quite sure though. Help is appreciated. ...