For questions about the writing of proofs. But instead of focusing the mathematics behind the proof, these questions ask about the details and implementation of the proof.

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0
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2answers
31 views

Let $a$ be an element of order $n$ in a group $G$. If $m$ and $n$ are relatively prime, then $a^m$ has order $n$.

Let $a$ be an element of order $n$ in a group $G$. If $m$ and $n$ are relatively prime, then $a^m$ has order $n$. Assume $m$ and $n$ are relatively prime, and that $a^m$ does not have order ...
-4
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0answers
67 views

Geometric Proof for Fermat's Last Theorem - A Question [closed]

I have been working on a geometric proof for Fermat's last theorem that I just realized has been worked on already in some shape or form (ba-dum-tsh). Before anyone says it, yes, I am aware that this ...
0
votes
1answer
33 views

Maximal solution of differential equation

Let $K\subset X$ be a compact set and let $x_0\in K$. Suppose that the maximal solution $x(t)$
0
votes
0answers
14 views

Show that $\mathbb{N}=\{ 0\} \sqcup S(n)$ and also that that union is disjoint?

Show that $\mathbb{N}=\{ 0\} \sqcup S(n)$ and also that that union is disjoint? I've been assigned this exercise in my lectures of elements of mathematics 2. Three axioms have been given for a Peano ...
0
votes
1answer
8 views

Euclidean algorithm to provde gcd's and multiples

Suppose a, b, n ∈ N. Use the Euclidean algorithm to prove that gcd(na, nb) = n gcd(a, b). I was going to try setting it up, by literally doing: nb = rna + k and so forth, but something tells me this ...
2
votes
1answer
187 views

two $\gcd$s that are coprime

Let $a, b$ and $c$ be integers. Prove that if $\gcd(a, b)$ and $\gcd(a, c)$ are coprime, then $\gcd(a, bc)$ = $\gcd(a, b) · \gcd(a, c)$ I am stumped in this problem. Can anybody clarify me what ...
0
votes
1answer
23 views

Proof: Probability using Induction

You have $n$ coins $C_1$, $C_2$, ..., $C_n$ for $n \in \mathbb{N}$. Each coin is weighted differently so that the probability that coin $C_i$ comes up heads is $\frac{1}{2i + 1}$. Prove by induction ...
-1
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2answers
32 views

prove the equivalence of the following statements: 2x-1 is irrational; x/3 is irrational

I am stumped. I really have no idea how to solve this problem. Can someone please help me through this? THE TWO EQUATIONS ARE SEPERATE
0
votes
2answers
60 views

Let $x, y \in\mathbb{R}$. Prove that $|x| + |y| \geq |x+y|$

I need to prove the following result: Let $x, y \in\mathbb{R}$. Prove that $|x| + |y| \geq |x+y|$ I know this is the triangle inequality, but I haven't seen one version that helps me solve this ...
1
vote
1answer
32 views

Let$\ \lim_{n\to \infty} \frac{ \ln n}{f(n)}=1$. If$\ a,b,c$ are natural, can we have$\ a^{b+c \ln n}\sim a^{c f(n)}$?

I shall note that$\ n$ as well goes through the natural numbers and that$\ f(n)$ is rational for any$\ n$. Also, I'm obviously excluding$\ a=1$. I'm inclined to think my claim is not possible ...
0
votes
1answer
19 views

Help complete proof that a matrix is orthogonal iif its column vectors are pairwise orthogonal and have length 1

Let ${\pmb{x}_1}$ and ${\pmb{x}_2}$ be vectors in $\mathbb{R}^2$ such that they are orthogonal and have length 1. Let $P$ be the 2x2 matrix whose column vectors are ${\pmb{x}_1}$ and ${\pmb{x}_2}$. ...
0
votes
0answers
24 views

Consider$\ f$, defined for all complex numbers except$\ x_0$. What can(not) happen to the real part of$\ f\left(x_0+bi\right)$ as$\ b\to0$?

Yes,$\ f$ is holomorphic. In other words, given that$$\ \lim_{x\to x_0}f(x)=\infty,$$what do we know a priori about $\ \lim_{b\to0}\Re\left(f\left(x_0+bi\right)\right)$? Can it be any extended real ...
1
vote
3answers
40 views

Prove the equivalence of the following 3 statements. [closed]

a) x is irrational b) 2x − 1 is irrational c) x/3 is irrational I am lost in this class I have no idea what to do.
1
vote
0answers
73 views

Is there a simple proof that $\int_0^1 \lceil f(x) \rceil \mathrm{d}x \geq \int_0^1 f(x) \mathrm{d}x$? [closed]

I want to prove that: $$\int_0^1 \lceil f(x) \rceil \mathrm{d}x \geq \int_0^1 f(x) \mathrm{d}x$$ Is there a simple proof that this is true? The purpose of this proof is for applying formal methods ...
0
votes
4answers
63 views

Is what I've done a proof? Proving there is always an rational number between two distinct rational numbers

The exercise I am working on is about proving whether there is always a rational number between two other distinct rational numbers. I came up with this $\frac{a}{b} < \frac{ad + bc}{2bd} < ...
1
vote
3answers
72 views

As$\ n \to \infty$, can a transcendental function$\ f\left(1+ \frac{1}{n}\right)$ to the power of$\ n$ tend to a rational power of$\ e$?

Let$\ f(n)$ be a transcendental function$\ \ne e^{g(n)}$, for any function$\ g(n)$. Does$$\ \lim_{n \to \infty} \left(f\left(1+ \frac{1}{n}\right)\right)^n =e^{ -k} = \lim_{n \to \infty} \left(1 - ...
0
votes
3answers
35 views

Use the Fundamental Theorem of Arithmetic to prove that if a>1 is composite, then there exists a prime p such that p|a and p≤√a

I know that since $a>1$ is composite, then it can be broken down into a product of prime factors, by Fundamental Theorem of Arithmetic. So then $a=p_1p_2\dots p_k$ for some natural number k. Then, ...
1
vote
3answers
44 views

show that every rational number has one and only one multiplicative inverse

I am stumped and have no idea on how I prove this. I don't know what else to say. I am beyond lost.
0
votes
1answer
51 views

Trouble understanding case analysis (proof by cases)

I've got a discrete math test coming up, and I've been studying religiously for the past week. Proof styles still frighten me though, I find it hard to wrap my head around them. Right now I am ...
1
vote
1answer
22 views

Difference between contradiction and paradox?

In multivalued logic one can distinguish at contradictions (of the type $P\wedge\neg P$) and paradoxes (of type $P\leftrightarrow \neg P$). How about in mathematics? Does the appearance of ...
1
vote
3answers
25 views

Can a limit of form$\ \frac{0}{0}$ be rational if the numerator is the difference of transcendental functions, and the denominator a polynomial one?

Let$\ f_1(x)$ and$\ f_2(x)$ be transcendental functions such that$\ \lim_{x\to 0} f_1(x)-f_2(x)=0$, and$\ f_3(x) $ polynomial, such that$\ f_3(0)=0$. Can$\ \lim_{x\to 0} \frac{f_1(x)-f_2(x)}{f_3(x)}$ ...
0
votes
2answers
59 views

Let$\ x$ be a real number between$\ 0$ and$\ 1$. Is it possible to write$\ e^{x}$ as a function of$\ \Gamma \left(x+1\right)$?

In particular, I'm looking for a relation between$\ e^x$ and$\ \frac{1}{ \Gamma \left(x+1\right) }$, which would be of help for a proof.
1
vote
2answers
31 views

Show that the intersections of the $G_s$ is normal subgroup of $G$

I need to prove that given a group $G$ acting in a set $S$, the intersection of the stabilizers $G_s$, where $G_s:=\{g\in G: g.s=s\}$ and $s$ varies through all $S$, is a normal subgroup of $G$. But ...
1
vote
1answer
54 views

Showing a set is a subset of another set

I need to show that $(A \cup B) \subseteq (A \cup B \cup C)$ My Work So Far: What I really need to show is that $x \in (A \cup B)$ implies $x \in (A \cup B \cup C)$ So I translated my sets into ...
0
votes
1answer
26 views

Suppose a, b, n ∈ N. Use the Euclidean algorithm to prove that gcd(na, nb) = n gcd(a, b).

For this problem, would I be able to say that by the properties of divisibility, if the GCD divides a and b, then it should also be able to divide any multiple n of a and b?
0
votes
2answers
39 views

How to verify by induction that 1(1!) + 2(2!) + … + n(n!) = (n+1)! - 1 for every pos. int. n?

Basis step: $n=1: 1(1!) = (1+1)! - 1 = 1$, true; $n=2 : 1(1!) + 2(2!) = 5 = (2+1)! - 1 = 6 - 1$, true; $n=3 : 1(1!) + 2(2!) + 3(3!) = 23 = (3+1)! - 1 = 24 - 1$, true; ... How do I prove the ...
1
vote
1answer
18 views

Order in direct proofs with even numbers

I'm doing an advanced maths class for high school and we have just started a topic about proofs. One of the questions (assume all numbers are integers here) is to prove that if $x\cdot y$ and $x + y$ ...
5
votes
4answers
617 views

Sum of cubes proof

Prove that for any natural number n the following equality holds: $$ (1+2+ \ldots + n)^2 = 1^3 + 2^3 + \ldots + n^3 $$ I think it has something to do with induction?
4
votes
1answer
71 views

Proof by counter example of optimal solution for Coin Changing problem (no nickels)

I'm a tutoring a student whose working on the classical coin changing problem. For those who are unfamiliar with problem or the greedy algorithm used for it. The goal is find the fewest number coins ...
2
votes
0answers
34 views

Applications of the Extended Euclidean Algorithm

I am asked to prove the statement: If $k$ is a common divisor of $a$ and $b$, then $k|gcd(a,b)$. I am also required to prove the converse. We can assume that $a, b, k$ are non-zero integers. I have ...
2
votes
2answers
68 views

Proving that if $p$ is a prime number then $gcd (p, (p-1)!) =1$

I am just making sure whether this is a valid proof: Since $p$ is a prime number, then $p$ is only divisible by $1$ or $p$ Suppose we want to take the $gcd (p,a)$ with a, an arbitrary ...
1
vote
1answer
19 views

Explanation of proof: if a graph $G$ has no isolated vertices and no even cycle, then every block of G is an edge of cycle

If a graph $G$ has no isolated vertices and no even cycle, then every block of G is an edge of cycle A block with 2 vertices is an edge. (Got it) A block $H$ with more than 2 vertices is ...
1
vote
3answers
68 views

proving that if $a, b$ are random non-zero integers, then $D$ is non-empty

Suppose a, b are two randomly chosen non-zero integers. Then the set $D = \{ax+by : x,y ∈ \mathbb{Z}, ax+by>0\}$ is non-empty. My lecturer wrote this up in my notes, saying that this is a ...
1
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3answers
54 views

Derivative proof

This proof is on derivatives. I have no idea where to even begin.
0
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2answers
30 views

Find integers $r$, $s$, and $t$ such that $12r + 30s + 18t = 2$

Could someone please explain if such integers exist and how to find them? If not, could someone please explain how to prove that they don't exist? Thank you!
-2
votes
1answer
50 views

Identify a countable union of nested intervals using the Archimedean principle [closed]

$\displaystyle\bigcup_{n=2}^\infty \left[\frac1n,3-\frac 2n\right]=(0,3)$. I can't prove using limits. I have to use the Archimedean principle and I don't know how to go about doing that..
1
vote
3answers
63 views

Proof by induction of a sum?

Let $n ∈ N$. Prove by induction that there are $n$ ways to write the number $n$ as a sum $n=x_1+x_2+...+x_k$ where the $x_i$ are natural numbers and $x_1 ≤x_2 ≤...≤x_k ≤x_1+1$. For example, $5 = 5$, ...
0
votes
1answer
47 views

Real Analysis Question: derivatives

Let $$f''(x)+p(x)\cdot f(x)=0$$ and $$g''(x)+p(x)\cdot g(x)=0$$ where $a<x<b$. 1 ) Show that $W=f'g-fg'$ is a constant on $(a,b)$. 2 ) Prove: If W$\neq$0 and $f(x_1)=f(x_2)=0$ where $a \lt x_1 ...
1
vote
1answer
33 views

Verification of identity $2\sum_{m,n\geq 1\,;\,m>n}\frac{1}{m^k n^k}=\left(\sum_{n\geq 1}\frac{1}{n^k}\right)^2 -\sum_{n\geq 1}\frac{1}{n^{2k}}$

Is this identity true? $$2\sum_{m,n\geq 1\,;\,m>n}\frac{1}{m^k n^k}=\left(\sum_{n\geq 1}\frac{1}{n^k}\right)^2 -\sum_{n\geq 1}\frac{1}{n^{2k}}$$ If so, how to prove it? Could you provide me a ...
7
votes
2answers
116 views

Prove that $\sum_{k=0}^\infty \frac{1}{16^k} \left(\frac{120k^2 + 151k + 47}{512k^4 + 1024k^3 + 712k^2 + 194k + 15}\right) = \pi$

How to prove the following identity $$\sum_{k=0}^\infty \frac{1}{16^k} \left(\frac{120k^2 + 151k + 47}{512k^4 + 1024k^3 + 712k^2 + 194k + 15}\right) = \pi$$ I am totally clueless in this one. Would ...
1
vote
4answers
270 views

Use the division algorithm to prove that 3|(n³ + 2n) for all n ∈ ℕ

I can do it by induction, thanks to the wonderful people of this website, but I'm not sure how to do it by the Division algorithm. Can anyone help me? I think I can show how 3 divides 2n, but I'm not ...
0
votes
3answers
37 views

Prove that if $a|b$ and $a|c$ then $a|(sb+tc)$ for all $s, t \in \mathbb{Z}$

Would this be the same thing as saying "Prove that if $a|b$ and $a|c$ then $a|(sb+tc)$ for any $s, t \in \mathbb{Z}$"? I can do the proof for any integers $s$ and $t$, but if any and all mean the same ...
2
votes
2answers
53 views

Prove or disprove that if a|(sb+tc) for all (and for some) s,t ∈ ℤ, then a|b, and a|c.

So, this is actually 2 questions in 1. I apologize if that is bad practice, but I didn't want to write 2 questions when they're a word different. So, I have Prove or disprove that if $a|(sb+tc), ...
2
votes
3answers
47 views

Show that $G_{s}$ is a normal subgroup of $G$

Definition: $G_{s}:=\{g \in G: g.s=s\}$ My attempt is the following: We take $g \in G$, and we consider this two sets: $$gG_{s}:=\{gh:h\in G_{s} \}$$ $$G_{s}g:=\{hg :h\in G_{s}\}$$ and we will ...
0
votes
1answer
22 views

Prove that a $\kappa : G/G_{s} \to G.s$ is a bijection

I have to prove that given an action this function $\kappa : G/G_{s} \to G.s$ is a bijection. $$ G/G_{s} \to G.s$$ $$gG_{s} \to g.s$$ Where $G$ is a group and: $G_{s}:=\{g \in G : g.s=s\}$(Isotropy ...
1
vote
1answer
21 views

Finding measure of skewness for binomial distribution

Here's how it was done in my class: $E[(X)_3]= n(n-1)(n-2) p^3$ (Calculated using definition. I understand that part properly.) $E[(X)_2]= n(n-1)p^2$ (Calculated using definition again). Now, ...
4
votes
1answer
57 views

Proving $f(x)$ attains $\max$ or $\min$ when $f(x)\to0$ as $|x|\to\infty$.

Suppose $f:\Bbb R\to\Bbb R$ is continuous such that $f(x)\to0$ as $|x|\to\infty$. Prove that $f$ attains either a maximum or a minimum. My attempt at the question : Given $\epsilon > 0 \ \ ...
1
vote
2answers
91 views

Proof Question using Proof By Contradiction, irrationality of $a + \sqrt[b]{5}$

I answered this question but am not quite sure if i did what was correct. If anything is wrong please point it out, thanks. Question:Prove that any number of the form $a + \sqrt[b]5$ is irrational, ...
0
votes
5answers
46 views

Prove by Induction $F_{2n} = F_{n} * L_{n}$, for n >= 1

Where $F$ is the Fibonacci Sequence, and $L$ is the Lucas Sequence. I need to find the inductive proof of this statement. I've got nearly a page of work in front of me trying to use definitions such ...
0
votes
0answers
34 views

Proof Verification/Alternative to Induction- Well ordering proof

This question grew out of Induction and Maximum Principle, which yours truly asked Sep 23, at 11:51. Due to Mauro Allegranza's suggest, I changed focus so as to first prove the equivalence of $(a)$ ...