# 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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### Can $\text{rank} (T) + \text{nullity} (T) = \dim V$ be proven with this simple argument?

I am helping one of my friends with linear algebra and gave him this theorem to prove as an exercise: Theorem . Let $V$ and $W$ be vector spaces over the field $F$ and let $T$ be a linear ...
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Prove: If $A_{n\times n}$ is not invertible then $Adjoint(A)$ is not invertible also. I have made the following: By contradiction. Suppose that $Adj(A)^{-1}$ exists Then write $Adj(A)*B=I$ \begin{...
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### $f$ is an entire function, then $f(\mathbb{C})$ is closed

While trying to prove "$f$ is an entire function such that $|f(z)| \to \infty, as |z|\to \infty$, then $f(\mathbb{C})$ is closed" I accidentally showed that "$f$ is an entire function, ...
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### Cantor's diagonal argument for proving the completeness of $L^\infty$ space, check work

I wrote up the following proof for proving $L^\infty$ is complete. Please check if I made any mistakes, thank you! There is a common/standard treatment for establishing the limit of a Cauchy sequence ...
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### Fermat's (hypothetical) erroneous proof

Until Wiles' proof of Fermat's last theorem all proposed proofs have been erroneous. It is not known which proof Fermat himself had in mind - but it is assumed that it was erroneous, too. Have there ...
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### Understanding the logic behind my "proof" of a false statement

I was working through these questions: (c) Prove that f(C1 ∩ C2) ⊆ f(C1) ∩ f(C2). (d) Give a counterexample to the opposite inclusion in (c). The opposite of c. (...
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### On a proof of the converse supporting hyperplane theorem

In connection with reading a proof of a theorem in a real-analysis textbook, I've stumbled upon the following theorem. Theorem. Suppose $C$ is closed with non-empty interior and has a supporting ...
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### $\displaystyle\lim_{n\to\infty} \frac{\sin \frac{1}{n}}{\frac{1}{n}} = 0$, where is my mistake?

$\displaystyle\lim_{n\to\infty} \frac{\sin \frac{1}{n}}{\frac{1}{n}} = 1$, but when I evaluate it as follows I get $0$ as the result: \begin{align} \lim_{n\to\infty} \frac{\sin \frac{1}{n}}{\frac{1}{n}...
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### Where have I gone wrong in computing the integral closure of $k[X, Y, Z]/(Y^3 + Y^2X^2 + YX^2 + X^3Z)$?

I was trying to follow the strategy in this answer to compute the integral closure of the domain $$R := k[X, Y, Z]/(Y^3 + Y^2X^2 + YX^2 + X^3Z)$$ where $k$ is a field. During the computation, I ...
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### A faulty "proof" regarding exactness of derived functors

I've been trying to wrap my head around derived functors and have come upon the following chain of arguments, which seems to yield the unreasonabel conclusion that all left derived functors are exact. ...
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### sin/cos = cos/sin

To begin with, let's note the following: $$\tan = \frac{\sin}{\cos} \\ \sin = \tan \cdot \cos \\ \cos = \frac{\sin}{\tan}$$ If we replace one of the functions in $\frac{\cos}{\sin}$ according to the ...
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### Confusion about proof statement by implication

I have a question regarding proving "if p is true then q is true". One way, is to show p $\implies$ q is a true statement, so then if p is true, then q is true. The other way is to prove it ...
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### Find the flaw in this mapping between the naturals and reals

I was studying Cantor's diagonal argument etc. I was testing the ideas and I thought of the following mapping between the naturals and the reals and I need some help to find the flaw in it. For ...
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### A suggested proof of the uncountability (sic) of rational numbers

[Note: I initially made the mistake of posting this over at MathOverflow, where it was promptly closed. There are however some comments there. Link. ] At MathOverflow, I found a very interesting ...
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### Show that $B_1(0)$ is a closed set in the space $C([0,1])$
Let be $(C([0,1]),\Vert \cdot\Vert_{\infty})$ the normed space of continuous functions, equipped with the supremum norm $\Vert\cdot\Vert=\sup\limits_{x\in[0,1]}|f(x)|$. Show that \$B_1(0):=\{f\in C([0,...