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Questions tagged [fake-proofs]

Seemingly flawless arguments are often presented to prove obvious fallacies (such as 0=1). This tag is the appropriate place to ask "Where is the proof wrong?" when encountering such proof.

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Writing a Fake Proof for Real Analysis

mathematics community! I'm teaching a course in Real Analysis soon, and one thing I wanted to include were a few "fake proofs" for my students to evaluate. The research I've done hasn't turned up any ...
3
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0answers
39 views

Bounding 2D Brownian Motion expected barrier hitting time of a square

I have been doing a computation but I am making a mistake somewhere and cannot figure out where. The question I have is: where am I making my mistake? I will run through my reasoning: Consider ...
17
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3answers
1k views

Where did I go wrong with my odd proof that $\frac{3dx}{3x} = \frac{5dx}{5x} \iff 3=5$?

I don't know where I went wrong, but it's interesting for me. Please check where my fault is! It is obvious that the below equation is correct: $$\frac{3dx}{3x}=\frac{5dx}{5x}$$ $$u=3x$$and$$v=5x$$ $...
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1answer
52 views

A direct way to an inequality : Ferrari's identities

I would like to submit a recent answer that I gave (and I have deleted) where someone tolds me that I was "total wrong" this is the following : Begin Prove that $$\frac{x^2+y^2+z^2}{2}\geq (\alpha\...
4
votes
3answers
69 views

Which one of these geometry proofs are incorrect?

One proof proves an angle to be 90º, and the other proof proves the very same angle to be 60º. I've looked over both proofs several times but I can't find what the error in either one of them is. And ...
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0answers
64 views

Apparent proof that $\pi = 0$? Can’t find error? [duplicate]

Start by setting P = Pi/i. (As I’m on mobile I can't properly format) ...
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1answer
90 views

Can an error be found in this proof of Gödel's incompleteness theorem?

Can you find a division by zero error in the following short proof of Gödel's incompleteness theorem? First a little background. $\text{G($a$)}$ returns the Gödel number of the formula $a$. As in ...
8
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1answer
135 views

ln(2) contradiction

$\ln2\approx.693$, according to my calculator. It can be written as the infinite sum $$1-\frac12+\frac13-\frac14+\frac15-\frac16+\frac17-\frac18+\frac19-\frac1{10}\dots$$ Rearranging this infinite ...
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0answers
56 views

False Proof Identification [duplicate]

I'm having some trouble identifying where specifically the following proof is incorrect: Statement: Given any positive integer $n$, every n people have the same name. "Proof:" We prove by induction. ...
10
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4answers
500 views

How does this proof of the regular dodecahedron's existence fail?

On Tim Gowers' webpage he has an example "proof" of the regular dodecahedron's existence which he claims contains a flaw. He writes Of course, I have not written the above proof in a totally ...
10
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9answers
161 views

Fake proof that $1$ is the solution of $x^2+x+1=0$

So I have this false proof and I am honestly confused why this is happening. Consider $x^2+x+1=0$, then $x+1=-x^2$. Now by simply dividing the equation by $x$ we get $x+1+1/x=0$. Substituting $x+1=-...
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2answers
703 views

Finding mistake in proof that all numbers are even [closed]

Claim: The numbers $0,1,2,3,\dots$ are all even. Proof: We use strong induction to prove the statement '$n$ is even' for $n=0,1,2,3,\dots$ Base case: $n=0$ is an even number, hence the ...
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4answers
679 views

Adding infinite and finite numbers: why doesn't 0=1?

Okay, so, $$\infty + 1 = \infty$$ subtract infinity from both sides. $$1=0$$ At first I thought, duh, $\infty \neq \infty+1$, but, now, I'm just more confused because my brother rephrased it in ...
0
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3answers
68 views

Finding a mistake in this 'proof' that $1 = 2$. [duplicate]

I have some idea of where the mistake in this 'proof' may be, but can't quite formalize. We start with the trivially correct statement, $1 - 3 = 4 - 6$. Then, completing the square on the LHS, we add ...
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2answers
56 views

About an exercise in Rudin's book

In the book of Walter Rudin Real And Complex Analysis page 31, exercise number 10 said: Suppose $\mu(X) < \infty$, $\{f_n\}$ a sequence of bounded complexes measurables functions on $X$ , and $f_n ...
0
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3answers
91 views

What is the correct proof that when $\epsilon$ decreases (in a limit) then the largest $\delta$ should also decrease?

In my search for this statement I wrote this question (a few years ago) When $\delta$ decreases should $\epsilon$ decrease? (In the definition of a limit when x approaches $a$ should $f(x)$ approach ...
3
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2answers
65 views

Uniform limit of functions with intermediate value property has intermediate value property? Where is the error in this proof?

This is the question that I posed myself and set out to solve before I came to MSE: Suppose that $\{f_n\}$ is a sequence of real-valued functions defined on $[a,b]$ such that each function has the ...
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3answers
81 views

Lesser-known fake proofs [closed]

I believe I have seen almost every elementary bogus proof of $0=1$ or $1=2.$ I am wondering if there are any lesser-known "proofs" where the error is harder to spot.
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3answers
200 views

$ \operatorname{dist}(w,A) \ge \operatorname{dist}(w, \partial A)?$ What is the mistake in this proof?

Suppose $(X,d)$ is a metric space, $w \in X$ amd $A \subseteq X$. $\partial A$ refers to the set of boundary points of the set $A$ defined as $\{ x \in X~|~ \operatorname{dist}(x,A)= 0 = \...
2
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1answer
36 views

Wrong proof for the variance of a sum of normally-distributed variables?

I'm reading the book "Introduction to Error Analysis" by John R. Taylor. The author is discussing the probability distribution of a sum of two normally-distributed random variables, and wants to show ...
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2answers
56 views

On The “Commutative” Properties Of Subtraction And Division

Can someone explain to me why subtraction and division is both commutative? The reason I believe that they are commutative are as follows: $$ 3-2 = 3+(-2) $$ $$ 3+(-2) = (-2)+(3) $$ I've read a ...
2
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4answers
185 views

How to prove that $-x$ is not equal to $x$ just because they yield the same result when in $x^2$

I know how incredibly stupid this sounds, but bear with me. Let's take any random $x$, say $3$, and any random $-x$, say $-3$. Let's plug it into $x^2$. They will both give the same result! I know ...
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0answers
93 views

Questions about the statement “Every number can be specified by less than twenty words.”

This is really an interesting question, though I do not know how to word it in a mathematical way. I am glad if one can help me to reword it mathematically. A friend of mine comes up with this ...
3
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1answer
30 views

Erroneous proof that derivative tending to $\infty$ implies that the function is not uniformly continuous.

I have a proof for the following statement (which I know is wrong): Let $f:(a,b)\rightarrow \mathbb{R}$ be a differntiable function such that $\underset{x\rightarrow a^+}{\lim} \vert f'(x) \vert=\...
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2answers
66 views

Proof that the set of Natural numbers are equal to the Real numbers [duplicate]

Obviously the proof is flawed in some way, but I have been struggling to find out where the flaw is. I am self taught and fairly new to set theory so the flaw may be obvious (sorry): The idea is that ...
4
votes
1answer
61 views

Topology of almost everywhere convergence - What's wrong with my construction?

I understand that the a.e. convergence is not topologizable, as there is contradiction, for instance with the fact that $L^p$ convergence does not imply a.e. convergence. So I am asking what is ...
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1answer
213 views

Is this considered a disproof of $1+2+3+4\ldots=-\frac{1}{12}$? [closed]

The series "$1+2+3+4\ldots=-\frac{1}{12}$" didn't seem to make sense to me as it breaks clear rules of series (but I am yet to research if the rules it broke aren't breakable(doesn't work if broken) ...
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3answers
89 views

Find all $x$ such that $4^{27}+4^{1000}+4^{x}$ is a perfect square.

Let,$4^{27}+4^{1000}+4^{x}=n^2$ $4^{2}(4^{25}+4^{998}+4^{x-2})=n^2$ LHS is multiple of 4.$n$ is also multiple of 4 Let $n=4a _1$. $4^{2}(4^{25}+4^{998}+4^{x-2})=16.a_1^2$ $4^{25}+4^{998}+4^{x-2}=...
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3answers
55 views

Bogus Proof by Strong Induction

So here is a bogus proof. Let $$P(n) ::= \forall k \leq n, a^k =1, $$ where k is a nonnegative valued variable. Base Case: $ P(0) $ is equivalent to $a^0 = 1$, which is true by definition of $...
0
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0answers
64 views

All positive integers $n$ are equal

Is my proof correct? I have to prove that: All positive integers $n$ are equal $P (n)$: All numbers in a set of $n$ positive integers are equal Base: $P (1)$ is true Inductive hypothesis: Suppose ...
3
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2answers
48 views

Misunderstanding Caley-Hamilton Theorem - Characteristic Polynomial in the Standard/Factorized Form

so my question is about the Caley-Hamilton theorem. Consider the following Matrix A. $$A =\begin{pmatrix} -1 & 0 & 4 \\ 2 & -1 & 0 \\ 3 & 2 & -1 \end{pmatrix}$$ The ...
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1answer
33 views

Contradiction in Proof

While studying Ahlfor's Complex Analysis text, I came upon this theorem: Theorem II. Suppose that $f(z)$ is analytic at $z_0$, $f(z_0) = w_0$, and that $f(z) - w_0$ has a zero of order $n$ at $...
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1answer
83 views

How can one trust geometric proofs if humans are susceptible to optical illusions?

How can one make a proof that doesn't consist of a bunch of symbolic manipulations "formal"?
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1answer
70 views

Why the wrong result?

$-\sqrt{-3} * \sqrt{-3} = x$ $-(-3^{\frac{1}{2}}) * (-3^{\frac{1}{2}}) = x$ $ +3^{\frac{1}{2}} * -3^{\frac{1}{2}} = x$ $-3^-1 = x$, then $x = -3$ The correct result should be $3$ positive And i ...
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0answers
31 views

Is this rough intuition for Brouwer's fixed point theorem in $\mathbb R^2$ correct?

To prove Brouwer's fixed point theorem on a closed disk: For any continuous function $f$ that maps a closed disk in $\mathbb R^2$ to itself, there exists a point $x_0$ in that disk, such that $f(...
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2answers
71 views

Does $(x^m)^n$ not always equal $x^{mn}$?

I was messing around and "proved" that $|x|=x$: $$|x|=\sqrt{x^2}=x^\frac{2}{2}=x^1=x.$$ Now clearly this cannot be true unless $x$ is non-negative. The error seems to be going from the second equality ...
3
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1answer
116 views

Paradoxical result from the chain rule

I noticed a very simple problem, yet paradoxical when I was solving a different problem. It would be great if you help me understand which of the following lines lead to the paradoxical result and why ...
0
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1answer
81 views

I can't spot the mistake. What's happening?

Let x and y be equal integers, x or y isn't equal to 0 or 1. We have the following: $x=y$ -subtract x to both sides $x-x=y-x$ $0=y-x$ -divide y²-x to both sides $\frac{0}{y^2-x}=\frac{y-x}{...
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7answers
3k views

The “assumption” in proof by induction

The second step in proof by induction is to: Prove that if the statement is true for some integer $n=k$, where $k\ge n_0$ then it is also true for the next larger integer, $n=k+1$ My question is ...
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1answer
56 views

¿Where does equality fail? [duplicate]

I know this: $\sqrt{x}^2 = |x|$, but $\sqrt{(-1)^2} = \sqrt{(-1)^2}$ $(-1)^\frac{2}{2} = \sqrt{-1 * -1}$ $(-1)^1 = \sqrt{1}$ $-1 = 1^2$, then $-1 = 1$ What step is wrong?
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0answers
50 views

A lack of understanding of convergence in distribution

I read somewhere that if we have $X_n \Rightarrow X$ and $Y_n \Rightarrow Y$ it does not implies $X_n + Y_n \Rightarrow X + Y$ (where $\Rightarrow$ means weak convergence/convergence in distribution). ...
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1answer
32 views

Fidning roots of an arbitrary polynomial with a specific pattern

Given an arbitrary field $F$ and an arbitrary element $a\neq0$ of that field, I am sked to find all the integers $n \geq 1$ such that $x+a$ is a root of $x^n+a^n$ in $F[x]$ I proved it as follows: ...
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4answers
85 views

What is the error on this prove that says $-1=1$

So $e^{i\phi}=\cos(\phi)+i\sin(\phi)$ and let $\phi=2\pi$ so $e^{i2\pi}=\cos(2\pi)+i\sin(2\pi)=1$ Now $$e^{i2\pi}=1$$ and by taking the square root of both sides $$\sqrt{e^{i2\pi}}=\sqrt{1}$$ since $\...
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4answers
29 views

Describe $x/(x+1) < 0$ with an interval

I am doing this exercise in a Calculus textbook. Describe all solutions $\frac{x}{x+1} < 0$ as an interval. I think I did successfully and got the solution $x \in (-1,0)$ by considering signs of $x$...
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2answers
84 views

$i=-i$ false proof

I was playing around with complex numbers and came upon this false proof. I don't see the mistake here. $$i^3=i^2i$$ $$i^3=-i$$ but $$i^3=i^{6/2}$$ $$i^3=√(i^6)$$ $$i^3=√((-1)^3)$$ $$i^3=√(-1)$$ $$i^...
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4answers
2k views

What's wrong with this “proof” that Gödel's first incompleteness theorem is wrong?

Edit: I've added an answer myself, based on the other answers and comments. Here is a very very informal "proof" (sketch) that Gödel's theorem is wrong (or at least that the idea of the proof is ...
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2answers
127 views

What is the parametrization used in the $\pi = 4$ proof?

You've probably seen this fallacious proof that $\pi = 4$: The answers to this question provide a variety of explanations for why it's fallacious. But I'd like to try a different approach. Let the ...
2
votes
2answers
73 views

right angle equal to obtuse triangle?

Given the obtuse angle x, we make a quadrilateral $ABCD$ with $∠DAB = x$, and $∠ABC = 90◦$, and $AD = BC$. Say the perpendicular bisector to $DC$ meets the perpendicular bisector to $AB$ at $Q$. Then $...
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votes
5answers
62 views

Where is the flaw in this proof that “any algebra variable does not equal any number”?

A friend of mine showed me this proof to demonstrate that any algebra variable does not equal any number. his proof relies on this idea $$0\neq1$$ $$0x\neq1x$$ $$0\neq1x$$ $$0\neq x$$ Full Proof: $$\...
1
vote
2answers
142 views

Visual Proof for Four colour theorem

I found a seemingly elegant, visual argument that shows why the four colour theorem (4CT) is true. The argument is as follows (I drew some pictures as well so I hope that helps): . Once again, I ...