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

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47
votes
15answers
15k views

Zero divided by zero must be equal to zero

What is wrong with the following argument (if you don't involve ring theory)? Proposition 1: $\frac{0}{0} = 0$ Proof: Suppose that $\frac{0}{0}$ is not equal to $0$ $\frac{0}{0}$ is not equal to $0 ...
45
votes
12answers
7k views

Proof that every number ≥ $8$ can be represented by a sum of fives and threes.

Can you check if my proof is right? Theorem. $\forall x\geq8, x$ can be represented by $5a + 3b$ where $a,b \in \mathbb{N}$. Base case(s): $x=8 = 3\cdot1 + 5\cdot1 \quad \checkmark\\ x=9 = 3\cdot3 ...
38
votes
12answers
5k views

Why do I get one extra wrong solution?

I'm trying to solve this equation: $$2-x=-\sqrt{x}$$ Multiply by (-1): $$\sqrt{x}=x-2$$ power of 2: $$x=\left(x-2\right)^2$$ then: $$x^2-5x+4=0$$ and that means: $$x=1, x=4$$ But $1$ is not a ...
38
votes
9answers
1k views

A game with $\delta$, $\epsilon$ and uniform continuity.

UPDATE: Bounty awarded, but it is still shady about what f) is. In Makarov's Selected Problems in Real Analysis there's this challenging problem: Describe the set of functions $f: \mathbb R ...
36
votes
15answers
4k views

Is it a new type of induction? (Infinitesimal induction) Is this even true?

Suppose we want to prove Euler's Formula with induction for all positive real numbers. At first this seems baffling, but an idea struck my mind today. Prove: $$e^{ix}=\cos x+i\sin x \ \ \ ...
34
votes
7answers
2k views

Find the value of $\sqrt{10\sqrt{10\sqrt{10…}}}$

I found a question that asked to find the limiting value of $$10\sqrt{10\sqrt{10\sqrt{10\sqrt{10\sqrt{...}}}}}$$If you make the substitution $x=10\sqrt{10\sqrt{10\sqrt{10\sqrt{10\sqrt{...}}}}}$ it ...
34
votes
3answers
2k views

Fascinating Lampshade Geometry

Today, I encountered a rather fascinating problem in a waiting room: Notice how the light is being cast on the wall? There is a curve that defines the boundary between light and shadow. In my ...
32
votes
1answer
2k views

Checking a Proof of a Theorem

Theorem 1.2 of Bennett and Skinner (Canad. J. Math., 2004) asserts that the Diophantine equation $x^{p} - 4y^{p} = z^{2}$ is unsolvable for every prime $p \geq 7.$ The following is a possible proof ...
29
votes
11answers
990 views

Is this $\gcd(0, 0) = 0$ a wrong belief in mathematics or it is true by convention?

I'm sorry to ask this question but it is important for me to know more about number theory. I'm confused how $0$ is not divided by itself and in Wolfram Alpha $\gcd(0, 0) = 0$ . My question here is: ...
27
votes
2answers
2k views

Lamport claims there is an error in Kelley's proof of the Schroeder-Bernstein theorem. What is it?

In section 4.1 of his note How to write a proof, Leslie Lamport mentions an error in Kelley's exposition of the Schroeder-Bernstein theorem: Some twenty years ago, I decided to write a proof of ...
25
votes
10answers
4k views

Proving $\sqrt 3$ is irrational.

There is a very simple proof by means of divisibility that $\sqrt 2$ is irrational. I have to prove that $\sqrt 3$ is irrational too, as a homework. I have done it as follows, ad absurdum: Suppose ...
25
votes
3answers
3k views

Can't find mistake in an easy proof.

Consider the following theorem. $\textbf{Theorem:}$ for any sets $A, B, C, D$, if $A \times B \subseteq C \times D$ then $A \subseteq C$ and $B \subseteq D$. Then the following proof is given. ...
24
votes
5answers
4k views

Proof of $(\mathbb{Z}/m\mathbb{Z}) \otimes_\mathbb{Z} (\mathbb{Z} / n \mathbb{Z}) \cong \mathbb{Z}/ \gcd(m,n)\mathbb{Z}$

I've just started to learn about the tensor product and I want to show: $$(\mathbb{Z}/m\mathbb{Z}) \otimes_\mathbb{Z} (\mathbb{Z} / n \mathbb{Z}) \cong \mathbb{Z}/ \gcd(m,n)\mathbb{Z}.$$ Can you tell ...
23
votes
10answers
13k views

Is a matrix $A$ with an eigenvalue of $0$ invertible?

Just wanted some input to see if my proof is satisfactory or if it needs some cleaning up. Here is what I have. Proof Suppose $A$ is square matrix and invertible and, for the sake of ...
23
votes
2answers
528 views

Would this solution of the limit of the sequence be correct?

Let's suppose that I have the sequence $a_n = \frac{1}{n^2} + \frac{2}{n^2} + \frac{3}{n^2} + \ldots + \frac{n}{n^2}, n \in \mathbb{N}$. And I have to find the limit of the sequence as $n \rightarrow ...
22
votes
5answers
4k views

Is a brute force method considered a proof?

Say we have some finite set, and some theory about a set, say "All elements of the finite set $X$ satisfy condition $Y$". If we let a computer check every single member of $X$ and conclude that the ...
22
votes
3answers
665 views

The series $\sum_{n=1}^\infty\frac1n$ diverges

We all know that the following harmonic series $$\sum_{n=1}^\infty\frac1n=\frac 1 1 + \frac 12 + \frac 13 + \cdots $$ diverges and grows very slowly!! I have seen many proofs of the result but ...
22
votes
1answer
1k views

Is this proof of the infinitude of primes valid?

The current issue (May 2015) of the American Mathematical Monthly has a one-line proof that there are an infinite number of primes, and I don't see why it is correct. Here is the proof: If the set ...
8
votes
0answers
273 views

A question on odd perfect numbers

Let $\sigma(x)$ be the sum of the divisors of the positive integer $x$. If $\sigma(M) = 2M$, then $M$ is said to be perfect. Currently, there are $48$ known examples of even perfect numbers -- on ...
8
votes
0answers
234 views

Direct proof that $\sqrt{2}$ is irrational? [duplicate]

Possible Duplicate: Irrationality proofs not by contradiction I've been puzzled for some days now, and I can't come up with an answer. I'm trying to come with a direct proof that $\sqrt{2}$ ...
7
votes
2answers
951 views

Proving a palindromic integer with an even number of digits is divisible by 11

I'm in an introductory course for discrete math so I'm a novice at English proofs. I'm not sure if my reasoning here is valid or if I'm using modular arithmetic correctly. Specifically the line I ...
6
votes
3answers
652 views

Proving $fg$ and $f+g$ is Riemann integrable through the easy and hard way.

Problem: Suppose $f,g$ are Riemann integrable functions, show that $f+g$ and $fg$ are also Riemann integrable. I know there is really easy to do this with measure theory, but I want to see if ...
5
votes
1answer
91 views

Munkres exercise

Problem Let $\{A_\alpha\}$ be a collection of subsets of $X$; let $X=\bigcup_{\alpha}A_\alpha$. Let $f:X\rightarrow Y$;suppose that $f\vert_{A_\alpha}$ is continuous for each $\alpha$. An index ...
5
votes
1answer
31 views

Where is the error in this proof of the Hodge theorem?

Let $(M,g)$ be a closed smooth Riemannian manifold. The following is the decomposition part of the Hodge theorem: Theorem The canonical map $\mathscr{H}^k(M)\to H^k(M)$ from harmonic $k$ ...
4
votes
1answer
62 views

Solving for a variable in an inverse function

I was asked to solve this formula for $R_2$: $$\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$$ So I did the following: \begin{align*} \frac{1}{R_2} &= \frac{1}{R} - ...
4
votes
0answers
25 views

Eigenvalues of $MA$ versus eigenvalues of $A$ for orthogonal projection $M$

Suppose that $M$ is symmetric idempotent $n\times n$ and has rank $n-k$. Suppose that $A$ is $n\times n$ and positive definite. Let $0<\nu_1\leq\nu_2\leq\ldots\nu_{n-k}$ be the nonzero ...
4
votes
0answers
33 views

Find all real $x$ such that $1990[x] +1989[-x]=1$ (where $[x]$ is the floor function for $x$).

Find all real $x$ such that $1990[x] +1989[-x]=1$ (where $[x]$ is the floor function for $x$). My effort Rearranging our equation we have : \begin{array}{c} 1990[x]+1989[-x]&=1 \\ ...
3
votes
4answers
72 views

If $B\subset A$ and $f:A\to B$ is injective prove it's a bijection between $A$ and $B$

I want to show that if $B\subset A$ and $f:A\to B$ is an injective function then there's a bijection between $A$ and $B$. I believe my "proof" is wrong, I probably use too much "intuition" when I ...
3
votes
1answer
61 views

Show that $\,f(x) = \sum_{n=0}^\infty a_nx^n$, for $x \in [0,1]$, is of bounded variation

Let $\{a_n\} \subset \mathbb{R}$, be such that $\sum_{n=0}^\infty \lvert a_n\rvert < \infty$. Define $$f(x) = \sum_{n=0}^\infty a_nx^n \quad \text{for } x \in [0,1]$$ Prove that $f$ is of bounded ...
3
votes
1answer
64 views
+50

Differential equation where Picard-Lindelöf can not be applied

My question is the following : Let $f:\mathbb{R}\to\mathbb{R}$ be continuous function and let $u:[a,b]\to\mathbb{R}$ be a $C^1$ function such that $$\forall t\in[a,b],u'(t)=f(u(t))\text{ and ...
2
votes
2answers
121 views

Proving either $x^2$ or $x^3$ is irrational if $x$ is irrational

I had a test today in discrete mathematics and I am dubious whether or not my proof is correct. Suppose $x$ is an irrational number. Prove that either $x^2$ or $x^3$ is irrational. My Answer: ...
2
votes
1answer
30 views

Geometrically interpreting complex numbers.

Prove that $|e^{i \alpha} - e^{i \beta}| |e^{i \gamma} - e^{i \delta}| + |e^{i \beta} - e^{i \gamma}| |e^{i \alpha} - e^{i \delta}| = |e^{i \alpha} - e^{i \gamma}| |e^{i \beta} - e^{i \delta}|$ ...
1
vote
2answers
33 views

prove limit of exponential function without concept of logarithm

The question is, prove that if a real number $x>1$, then $\lim_{n\to\infty}x^n = \infty$, where $n \in \mathbb N$, without using the logarithmic concept. I came up with a proof, but I'm not so sure ...
1
vote
1answer
17 views

How to convert sentence into logic formula

Hi I wanted to know if I have converted this sentence into propositional logic correctly. This is the sentence At least two of the propositions $p$, $q$ and $r$ are true. and this is my answer ...
1
vote
1answer
52 views

Hartshorne Exercise II.2.18(d)

The Exercise: Let $\phi: A \rightarrow B$ be a ring homomorphism and let $X = \operatorname{Spec} A, Y = \operatorname{Spec} B$. Let $f: Y \rightarrow X$ be the morphism of schemes induced by $\phi$. ...
1
vote
2answers
590 views

Proving that hyperbolic sinh is bijective

I have to prove that sinh is bijective. So first i try to profe that it is 1-1: $f(a)= f(b) => a=b$ I will use proof by contradiction: let f(a)= f(b) and $a$ doesn't equal to $b$. After i ...
1
vote
3answers
38 views

What is the significance of using prime numbers in proving: $x$ is a multiply of $y$?

I came to a problem where it asks me to prove, for example, $n^4-n^2$ is a multiple of $12$. Now, factorize the multiple: $n\times n\times (n-1)\times (n+1)$. Here we have $3$ consecutive integers. ...
1
vote
1answer
43 views

Proving the volume of sphere by using tiny volumes

How can I prove the volume of sphere, by using many cones starting at the center of the sphere? It doesn't have to be cones, pyramids also work.
1
vote
0answers
20 views

Linear transformation representation proof

Hi I am wanting for someone to go over what I have and possibly correct my mistakes. Or any comments on the techniques, etc. I want to prove that if $V$ and $W$ are vector spaces over some field F, ...
0
votes
1answer
29 views

Real Analysis, Folland 3.4.26, Differentiation on Euclidean Space

Background Information - A Borel measure $\nu$ on $\mathbb{R}^n$ will be called regular if i.) $\nu(K) < \infty$ for every compact $K$ ii.) $\nu (E) = \inf\{\nu(U): E\subset U, U \ ...
0
votes
1answer
59 views

Is there anything wrong in the following proof?

Problem. Let $(X,d_X)$ and $(Y,d_Y)$ be two metric spaces and let $U\subseteq X$ and $V\subseteq Y$ such that $U$ and $V$ are respectively open in $X$ and $Y$. Show that $U\times V$ is open in ...
0
votes
1answer
45 views

Detail in proof $(A \cup B)'=A'\cup B'$

Let $(X,d)$ be a metric space. Given $A\subset X%$, denote by $A'$ be the set of all limit points of $A$ in $X$. Now, I know several ways to prove that $(A\cup B)'=A'\cup B'$ which I'm perfectly ...
0
votes
1answer
53 views

Is every finite subgroup of multiplicative subgroup of an integral domain cyclic?

Here is a well-known theorem: Let $F$ be a field and $G$ be a finite subgroup of $F^*$. Then, $G$ is cyclic. Can this theorem be extended to integral domain? Here's a lemma in Weil's elementary ...
0
votes
1answer
36 views

Discrete math 4 proofs

I have a few questions that i have answered but i am not sure if its proof enough, if you could help me out and tell me if i am correct or not thta would be great. Question 1: For all sets A,B,C we ...
0
votes
0answers
17 views

Connected sets prove that definitions are equivalent

I found the following two definitions of connected set. I couldn't really see how they were equivalent so I tried to prove it. Definition: Two subsets $A$ and $B$ of a metric space $X$ are said to ...
0
votes
0answers
10 views

A graph $G=(V,E)$ is connected and a vertex $s \in V$ is not a vertex separator iff $G-s$ is connected - requires $deg(s) > 3$?

I was asked to prove that $G$ is connected and $s$ is not a vertex separator iff $G-s$ is connected, given that $deg(s) > 3$. I'm struggling to understand why I need the $deg(s) > 3$ part. One ...
0
votes
0answers
36 views

Probability that a given function is prime…

If we have a set of primes $p_1$, $p_2$, ... , $p_n$, we can easily construct a function of their product: $$f(\alpha) = \alpha \left( \prod_{k=1}^n{p_k} \right) + 1, \alpha \in \mathbb{N}$$ I'm ...
0
votes
0answers
29 views

How to calculate the length of this plane curve (loxodrome/rhumb line)?

I am trying to calculate the length of a (what I believe is) a loxodrome, using differential geometry. I am given a curve $\gamma(t)=\big(\theta(t),\varphi(t)\big)\subset \mathbb S^2$ that ...
0
votes
0answers
64 views

Is it correct? $1^n +2^n +…+(p-1)^n=-1 \pmod p$

$p$ a prime number, $n\in \mathbb{N} $ and $p-1\mid n$ then $1^n +2^n +...+(p-1)^n=-1 \pmod p$ I'm not sure if my proof is correct: Take the group $G=(\mathbb{Z}^{*}_{p},\cdot)$ with the ...
0
votes
0answers
76 views

Prove that each diagonal of a quadrilateral lies either entirely in its interior or entirely in its exterior.

From Kiselev's Planimetry problem 55. Here's my attempt at a proof, I'd appreciate feedback on how to get better beyond whether or not it's correct, as I have very little experience and am still ...