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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4answers
102 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} < ...
0
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3answers
87 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
47 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
66 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
167 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 ...
2
votes
1answer
37 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
28 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
65 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
37 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 ...
2
votes
1answer
68 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
2answers
86 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
48 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 ...
2
votes
1answer
22 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 $x + y$ ...
4
votes
4answers
2k views

Sum of cubes proof [duplicate]

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
275 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
2answers
88 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
53 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
82 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
vote
3answers
60 views

Derivative proof

This proof is on derivatives. I have no idea where to even begin.
0
votes
2answers
38 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!
-1
votes
1answer
65 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
72 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
63 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 ...
2
votes
1answer
53 views

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

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
152 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
419 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
102 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
68 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
48 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
25 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
115 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, ...
5
votes
1answer
117 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
238 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
49 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 ...
1
vote
0answers
47 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 $(a)$ ...
1
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2answers
62 views

Can this be solve using modular arithmetic? $k$ is prime $\Rightarrow$ $8k+1$ is prime

Is the following statement true or false? $\forall k \in \mathbb{N}, k$ is prime $\Rightarrow$ $8k+1$ is prime The answer is that the statement is false because if $k=7$, then $k$ is prime but ...
-1
votes
2answers
37 views

Use division algorithm and then induction to show 3|(n³+2n) for all ℕ. [duplicate]

For division algorithm, would I do something along the lines of n³+2n = 3q+r and go from there? For induction, I did the base case, which is true, and so then I moved on to the k+1 case, in which I ...
1
vote
1answer
83 views

Prove that if $p\ge 5$ is prime, then $p^2 + 1$ is composite

So, coming off of this question, I know how to find out what the remainder is, so after figuring whether the remainder is $1$ or $5$, would I just plug in $p = 6q + (1\ \text{or}\ 5)$ into $p^2+1$? ...
1
vote
5answers
83 views

Proving by induction $5^{3n} + 2 \cdot 5^{2n} - 5^{n} - 2$ is divisible by $4$

I want to prove the following twice. Once by induction then again by any other method. $$5^{3n} + 2 \cdot 5^{2n} - 5^{n} - 2$$ is a multiple of 4 for all nonnegative integers n. Let n=0 , since it is ...
1
vote
2answers
467 views

Prove that the equation $x^{3}-3x+b=0$ has at most one root in the interval $[-1,1]$

I have to prove that the equation $x^{3}-3x+b=0$ has at most one root in the interval $[-1,1]$. My attempt: We consider the function $g(x)=x^{3}-3x+b$.Now since it is a polynomial it is ...
1
vote
1answer
47 views

Divisibility and Primes

Suppose that $p,q,r$ are prime numbers and $p$ is odd. If $p\,|\,(2q+r)$ and $p\,|\,(2q-r)$, prove that $q=r$. So I'm trying to use the definition of greatest common divisor to come up with two ...
0
votes
3answers
49 views

Find the characteristic of the ring $\mathbb Z_6 \times \mathbb Z_{15}$

My attempt: Let the characteristic be $n$. Then, $n \cdot (1_6, 1_{15}) = (0_6, 0_{15})$, i.e. $n \cdot 1_6=0_6$ and $n \cdot 1_{15}=0_{15}$ The least $n$ for which both are true is $30$, so $30$ ...
3
votes
5answers
110 views

How to select the right modulus to prove that there do not exist integers $a$ and $b$ such that $a^2+b^2=1234567$?

I understand the solution but I don't know how the author decided to start with modulo 4 instead of something else? What is it about the expression $a^2+b2=1234567$ that would trigger us to select ...
0
votes
2answers
36 views

Help me understand the proof of $a \equiv b \mod m \Rightarrow r_m(a)=r_m(b)$

Let: $r_m:\mathbb{Z}\rightarrow R_m$ where $r_m(a)=r\Leftrightarrow a \equiv r \mod m $ and $r\in R_m$ where $R_m$ is the set of residues modulo $m$. I understand the above proof until $r' ...
0
votes
1answer
98 views

Proof involving the Pigeonhole principle

Prove that among any given $n + 1$ positive integers, there are always two whose difference is divisible by $n$ My Answer: Using Pigeonhole principle: From a set of at least $2$ different $n+1$ ...
1
vote
1answer
62 views

What's the correct way of concluding an induction proof?

I had to prove that for every set $s$, the number of subsets with odd cardinalities is $2^{n-1}$. I concluded that this formula holds everytime $|s| \geq 1$ and then I used an inductive process to ...
0
votes
2answers
64 views

Continuity Question in Analysis

Prove that if $$f: A \to \mathbb R$$ is continuous at $a$ and $f(a) > 0$ then there exists $\delta > 0$ such that $$x \in (a - \delta, a + \delta )\cap A \implies f(x) > 0$$ Literally ...
0
votes
1answer
48 views

Linear Algebra: Projection of a Linear Transformation

I am having some confusion of this problem: Let $T:\mathbb{R}^3\to \mathbb{R}^3$. If $T(a,b,c)=(a,b,0)$, show that T is the projection on the xy-plane to the z axis. The following is the ...
1
vote
4answers
85 views

Can anyone explain how to show that $n^{5} -n ≡0$ mod $30$ for every $ n \in \mathbb{N} $

I first tried to answer this using proof by induction, however my problem got more complicated when I got to the induction step. Is there another way of solving this problem?
0
votes
1answer
63 views

If a theorem says “$A \iff B$” and I want to prove $A$, does it suffice to show $A \implies B$?

For example, if there is theorem that says: "$[x] = [y] \iff x \sim y$," and I am asked to prove $[(a,b)] = [(c,d)]$ Is it enough to show that $[(a,b)] = [(c,d)] \implies (a,b) \sim (c,d)$, because ...