Questions tagged [fake-proofs]

Seemingly flawless arguments are often presented to prove obvious fallacies (such as 0=1). This is the appropriate tag to use when asking "Where is the proof wrong?" about proofs of such obvious fallacies.

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Where is my mistake in showing $\nabla \mathbf{a}^{T}\mathbf{x}=a_{1}+a_{2}+\cdots+a_{n}$?

Let $f:=\mathbf{a}^{T}\mathbf{x}$. The claim that: $$\nabla f=\nabla\mathbf{a}^{T}\mathbf{x}=\|\mathbf{a}\|_{1}=a_{1}+a_{2}+\cdots+a_{n}\tag{1}$$ is false where in fact the true answer is $\mathbf{a}^{...
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2 votes
2 answers
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Question about demonstration of the type $A\Leftrightarrow B$

I have an important question when it is asked to us to proove affirmation/theorem wich look likes this $A\Leftrightarrow B$ I know that in order to proove $A\Leftrightarrow B$ i must proove first that:...
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4 votes
1 answer
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A Collection of Bogus Proofs

Hello M.S.E. people, This question is just for fun, don't take it seriously :). We have all encountered Bogus Proofs, which seem logical and reasonable, but they prove some claims which are completely ...
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26 votes
1 answer
755 views

Historical Mistake of Assuming Measurability

I am recently reading about Fourier transforms and convolutions. It was a surprise to me that it takes quite several paragraphs to prove the measurability of innocent looking $f(x-y)$ (reference: ...
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1 answer
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Do 2 of the same circles contain each other? (Proving / Disproving)

Two Circles, $C_1$ and $C_2$ have the same center $(h, k)$ and radius $r$. Trying to prove or disprove that $C_1$ contains $C_2$ and $C_2$ contains $C_1$ OR neither contains the other.. but I think I'...
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1 vote
0 answers
83 views

Is consistency with the $\omega$-rule absolute to $\omega$-models?

According to Wikipedia, a theory $T$ that interprets arithmetic is consistent with the $\omega$-rule if and only if it has an $\omega$-model. That would mean that consistency with the $\omega$-rule is ...
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6 votes
1 answer
405 views

False proof: Every linear operator (matrix) has an eigenvalue.

Below is a proof that any linear operator must have an eigenvalue. The proof obviously contains a mistake because the statement is wrong. But I do not see the mistake. Please point it out if you see ...
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0 votes
1 answer
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Error in a bogus proof showing equivalence of Cartesian Product and Union of sets of pairs

This question is related to another answered question, but I am struggling to relate the two (as a self-learner, I have no other avenues to get some feedback). The problem which I am trying is the one ...
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6 votes
1 answer
60 views

Where does this "proof" that $\Bbb{R}^\omega$ is normal in the box topology go awry?

James Munkres' Topology, 2nd Edition indicates that the space $$\Bbb{R}^\omega := \{ (x_0, x_1, x_2, ...) | x_i \in \Bbb{R}, \forall i < \omega \}$$ equipped with the box topology is completely ...
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1 answer
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What, if anything, is wrong with this condensed proof in Daniel Solow

What, if anything, is wrong with the following condensed proof? enter image description here
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1 vote
3 answers
81 views

Why doesn't$\frac{\sin a+\sin3a+\sin5a}{\cos a+\cos3a+\cos5a}=\tan3a$ imply $\sin a+\sin5a=0$ and $\cos a+\cos5a=0$?

$$\frac{\sin(\alpha)+\sin(3\alpha)+\sin(5\alpha)}{\cos(\alpha)+\cos(3\alpha)+\cos(5\alpha)} = \tan(3\alpha) \tag1$$ I've proven this trigonometric identity by subtracting the RHS from both sides and ...
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1 vote
0 answers
27 views

A possible proof about properties of vector spaces

I have the following proposition to prove: Proposition (1) Let $A:=\{\vec{v}_1,...,\vec{v}_n\}$ be a subset of an $F$-vectorial space $V$. If $\text{span}(A)=V$ then every subset of $V$ independient ...
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0 votes
1 answer
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Brackets in exponentiation [duplicate]

$ (-1)^{3}=((-1)^2)^\frac {3}{2}=(1)^\frac {3}{2}=1$ See, I know it's wrong but don't know why . Which rule is disobeyed here? And which rule should be followed in these cases?
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1 answer
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Fake Proof: Integral over Closed Contour in Complex Plane is Always Zero

I am trying to learn complex analysis, and want to understand how to evaluate contour integrals without using the residue theorem. But I don't understand how an integral over a closed contour can be ...
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9 votes
6 answers
1k views

Having trouble debunking my friend's "proof" that "There is no real number greater than 0"

My friend and I were talking about math stuff as usual, when he brought up a fake proof for the statement: There is no real number greater than $0$. Now obviously this isn't true, because any ...
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1 answer
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Proof That $\sin{(bt)} = b\sin{(t)}$?

Note the Taylor expansion for $\sin{x}$: $$ \sin{x} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots $$ Now consider the Taylor expansion for $\sin(bt)$: $$ \begin{align} \sin{(bt)} ...
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2 votes
1 answer
67 views

(Fake proof) Bounded linear operators $\mathcal{L}(X,Y)$ between separable Banach spaces $X$, $Y$ is itself a separable Banach space?

Fake proof: Let $X$ and $Y$ be separable Banach spaces. We know that the space $\mathcal{L}(X,Y)$ of bounded operators $X \to Y$ endowed with the operator norm is itself a Banach space. (Cf. [1][2][3]....
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1 vote
1 answer
47 views

Is my proof valid? Basic proof for set theory, also is the argument valid if the goal is the other way around?

Suppose $\mathcal{F}$ and $\mathcal{G}$ are nonempty families of sets. Prove that if $\mathcal{F} \subseteq \mathcal{G}$ then $ \bigcap \mathcal{G} \subseteq \bigcap \mathcal{F}$ Proof. Let $x$ and $...
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3 votes
1 answer
140 views

What's wrong with this proof of $0! = 1$

I was trying to find $0!$ using the series expansion of $e^x$ and got some odd results. We know that, $$e^x = \sum_{n =0}^\infty\dfrac{x^n}{n!}$$ Putting $x = 0$ will give us following results. $$e^...
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2 votes
0 answers
53 views

If dividing by a variable gives a contradiction, does that imply that the variable must be zero?

Solving a problem, I got to a situation of contradiction after a lot of algebra and, looking back, I divided both sides of the equation by a variable that could be zero. I then assumed that the ...
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2 votes
1 answer
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I "proved" that Cauchy-ness is preserved by continuous functions. Where did I go wrong?

Let $X_1$, $X_2$ be metric spaces and $f\colon X_1\to X_2$ be continuous. Let $(x_i)_i$ be Cauchy in $X_1$. I argue that $(f(x_i))_i$ is Cauchy in $X_2$: Let $\epsilon > 0$. Then there exists an $N$...
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1 vote
0 answers
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What are limitations of proofs by contrapositive?

If P then Q. Then we can say if not Q then not P, right? Now we will say that P is an even number and Q is a number such when it's multiplied by itself the result is a positive number. We will use ...
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0 votes
0 answers
232 views

How to know when the Collatz conjecture has been proved?

When using google to find out about research results about the Collatz conjecture, I find numerous proofs by various people who seem to be experts of the topic and an abundance of proofs by amateurs. ...
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0 votes
4 answers
73 views

Two solutions to complex equation causing a problem

If we look at the equation \begin{align} z = \sqrt{ 8 - 6 i }, \end{align} we will find the solutions \begin{align} z_1 = -3+i \end{align} \begin{align} z_2 = 3 - i \end{...
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0 votes
2 answers
92 views

$A\notin B$ set thoery

Let $A$ and $B$ be sets. Is true or false that $A\notin B$? I think It's false. Proof Suppouse that $A\notin B$. Then for any set $A$ and $B$ is the case that $A\notin B$, this implies that $\emptyset\...
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4 votes
0 answers
66 views

Is the category of measurable spaces MONOIDAL closed? (fake-proof)

As discussed on MathOverflow, there is a preprint on ArXiv that claims the category of measurable spaces is monoidal closed (in particular that exponential objects $Y^X$ always exist and have ...
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1 vote
1 answer
69 views

Fake Proof for Dimension of $SO(n)$ (rotations)?

Because there are $\binom{n}{2}$ distinct planes spanned by the elements of any given orthonormal basis for $\mathbb{R}^n$, it seems to follow readily (because apparently planar rotations generate all ...
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7 votes
2 answers
378 views

Where am I wrong that I can prove every subgroup is normal?

I appreciate helps for figuring out the problem of the following argument. I know it's certainly not the case that every subgroup of a group is normal. But I cannot find out where is this wrong. Let $...
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1 vote
1 answer
102 views

Fake proof that $0=1$ using $\int\frac{1}{x\ln x}\,\mathrm{d}x$ and integration by parts

$\newcommand{\d}{\mathrm{d}}$I saw, in a series of joke proofs that Santa Claus exists, the following: We want to evaluate the indefinite integral of: $$\int\color{green}{\frac{1}{\ln x}}\cdot\color{...
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0 votes
2 answers
103 views

Exercise $6.5$ of Baby Rudin

Suppose $f$ is a bounded real function on $[a, b]$, and $f^2 \in \mathscr{R}$ on $[a, b]$. Does it follow that $f \in \mathscr{R}$? Does the answer change if we assume that $f^3 \in \mathscr{R}$? ...
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1 vote
1 answer
113 views

Fake proof for a sequence convergence to a limit

Where does a false proof of a sequence breaks down. For example the sequence: $$\lim_{} \left(\frac{n+1}{n}\right)=1$$ If I want to prove that a sequence converges to a limit other than 1, take $0$. ...
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-3 votes
1 answer
47 views

Trouble identifying error in fake-proof. [duplicate]

I was talking to a friend about the problem with this proof and I'm stomped on the illegal step: $i = (-1)^{(1/2)}= (-1)^{(2/4)} =((-1)^2)^{(1/4)} = 1^{(1/4)} = 1$
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2 votes
2 answers
115 views

Question about the limit $\displaystyle\lim_{n\rightarrow \infty}\sqrt[n]{3^n+ 2^n}$

I have one question about limits: it is required to find the limit $\displaystyle\lim_{n \rightarrow \infty}\sqrt[n]{3^n + 2^n}$. I calculated it like this: $$(3^n+2^n)^{1/n} = \left(3^n \cdot \frac{3^...
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  • 388
0 votes
1 answer
48 views

Limits of $ f(x,y) = y\ (1-x)^{y-2} $ reach contradiction

Introduction I evaluated a limit of a multivariable function at two values and combining the two results, gives a contradiction. I want to understand where my mistake is: the limit evaluations or the ...
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  • 406
3 votes
1 answer
109 views

Detect Wrong Proof by strong induction $a^{n-1}=1$ for all $n$

Obviously there is some error in the steps, but i can't figure it out, appreciate some hint. We show using strong induction that $$ a^{n-1}=1 \hspace{1cm} \forall n\in \mathbb{N}$$ $$ n=1 \hspace{.5cm}...
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0 answers
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Question on non-continuous derivative

Let's assume that $f:\mathbb{R}\to\mathbb{R}$ is differentiable and its derivative $f'(x)$ is not continuous. Applying the mean value theorem we get: $$ \frac{f(x)-f(a)}{x-a}=f'(\xi), \text{ where } a&...
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1 vote
0 answers
58 views

In a topological group all loops based at 1 are homotopic

Let G be a topological group and $f,g : [0,1] \times [0,1] \to G$ two loops based at 1. Why isn't the application $$ H:\begin{array}{rcl} I \times I & \longrightarrow & G \\ (t,u) & \...
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0 votes
0 answers
6 views

Can we distribute exponents into the product (multiplication) of negatives? [duplicate]

This is hopefully a quick and easy answer, but I've found myself embarrassingly confused by what seems to be a simple statement. Written as $$ 2 = \sqrt{4} = \sqrt{(-2)(-2)} = (-2)^{1/2}(-2)^{1/2} = (-...
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3 votes
2 answers
100 views

Erroneous Proof that the Derivative is Continuous

I have written the following which argues that for any arbitrary sequence of points such that $x_n \to c, x_n\neq c$ we must also have $f'(x_n)\to f'(c)$ which would imply that $f'(x)\to f'(c)$ as $x\...
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0 votes
2 answers
109 views

Why is this proof wrong? $2^i = \frac{\sqrt{2}}{2}$ [closed]

If $i=-1^\frac{1}{2}$, then $2^i$ could be written as $2^{{-1}^{1/2}}$. By the multiplicative law of exponents, this should be equal to $2^{-1/2}$, which is $\frac{\sqrt2}{2}$. This doesn't feel right....
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0 votes
1 answer
61 views

There exists non Borel-measurable set in $[0,1]$

Show that not all subsets of $[0, 1]$ are Borel-measurable. Find a description of a non-measurable subset ? Hint: use the result about non-existence of shift-invariant measures on $([0, 1],\mathcal P([...
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0 votes
0 answers
41 views

An incomplete proof of d'Alembert's lemma from Stillwell's Elements of Algebra

I'm studying Stillwell's Elements of Algebra. In Section 3.8, he gives states d'Alembert's lemma and provides the following proof thereof with the aim of proving the fundamental theorem of algebra. d'...
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7 votes
3 answers
560 views

Please can someone tell accurately and precisely the actual fallacies of this fake proof that 0.999... ≠ 1.000...?

Please can someone tell accurately and precisely the actual fallacies of this fake proof that 0.999... ≠ 1.000... ? Define $F: \{\text{decimals in } [0,1]\} \to \{\text{decimals in }[0,1]\}$ by $a....
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0 votes
0 answers
63 views

Is this proof true for 0! = 1 [duplicate]

I see proof below for 0! = 1. Is this true in mathematical scientists' idea? (n+1)! = (n+1)n! → (n+1)! ÷ (n+1) = n! Now if we put 0 to n, we will have: 1! ÷ 1 = ...
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-4 votes
1 answer
105 views

Proof that $i=1=-1$( Mathematical Fallacy) [duplicate]

I thought of a proof that:- $$i=-1=1$$ Here is the proof :- $$ (i)^4=(i^2)^2 \\ = (\sqrt{-1})^4=(\sqrt{-1}^2)^2 \\ = (\sqrt{-1})^4=(-1)^2 \\ = (\sqrt{-1})^4=1 \\ \implies i^4=1 \\ \text{also} \\ -1^...
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1 vote
3 answers
109 views

Where is the error in this "proof" that $1=0$?

Suppose $$x-2z = 0$$ Here is what I did to the equation, by dividing $x-2z$ on both sides, $$\frac{x-2z}{x-2z} = \frac{0}{x-2z} \implies 1 = 0$$ I don't understand where I am wrong.
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0 votes
2 answers
109 views

Is there an equivalence between $a$ and $b$ when $ab \in H$, $H \subset G$

$$ab \in H \iff a \equiv b \pmod H$$ It abides by reflexivity, symmetry and transitivity. Proof of reflexivity: (taken from this site) $H \subset G \\ a,e \in G \\ e = aa^{-1} \in H$ Proof of symmetry:...
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3 votes
3 answers
86 views

$A\subset (B\cup C)$ $\implies$ $A\subset B $ or $A\subset C$

Clearly, it is not true that $A\subset (B\cup C)$ implies $A\subset B $ or $A\subset C$, since if $A=\{b,c\}, B=\{a,b\}, \text{ and }C=\{c,d\}$, then $A\subset B\cup C =\{a,b,c,d\}$ but $A\not\subset ...
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-4 votes
3 answers
149 views

Questions about the proof: "all positive integers are equal" [duplicate]

Others have expressed confusion about this proof on Stack and I have looked through every one of them because I don't want to post a duplicate post, however, none of them answer the questions I have ...
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1 vote
0 answers
67 views

Muirhead inequality doesn't require majorisation condition?

I just read the proof of Muirhead Inequality here and I thought it was really cool. But I do have a question, in that proof, nowhere was the fact that ${[a_i]}_{i=1}^n$ majorizes ${[b_i]}_{i=1}^n$ ...
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