Questions tagged [alternative-proof]

If you already have a proof for some result but want to ask for a different proof (using different methods).

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Simple Direct Proof of Nakayama's Lemma

Is there a simple direct proof of (this version of) Nakayama's Lemma? By simplicity, I am mostly referring to the concepts required to understand the proof, not how terse the proof is. I have no doubt ...
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Detailed Proof of Proposition 2.12 a) from Diamond, Darmon, Taylor, "Fermat's Last Theorem."

I seek a highly detailed proof of this statement in the split multiplicative reduction case. The result can be found on page 57 here. That is, if $E/\mathbb Q$ has split multiplicative reduction at $p$...
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Is there a brute force proof that the first-order characterization of a Jacobson radical is an ideal?

Let rings be commutative and unital. Is there a brute force proof that the first-order characterization of a Jacobson radical is an ideal? Basically, for context, I'm having some trouble wrapping my ...
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Problem 237 "Mathematical Quickies:270 Stimulating Problems with Solutions" Particle Movement

I was solving "Mathematical Quickies:270 Stimulating Problems with Solutions" when I came across a very peculiar question (Problem 237): A particle moves in a straight line starting from ...
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Show that $\frac{x^3+y^3}{x^2+y^2}$ is not differentiable at $(0,0)$

Let be $f:\mathbb{R}^2\to\mathbb{R}$ where $$f(x,y):=\begin{cases} \frac{x^3+y^3}{x^2+y^2},&(x,y)\neq (0,0)\\0,&(x,y)=(0,0).\end{cases}$$ Show that $f$ is not differentiable at $(0,0)$. My ...
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Using Generating Functions To Count Things

At the theater, the movie X is currently showing. However, there are only $7$ tickets left in row $F$, $8$ tickets left in row $G$, and $9$ tickets left in row $H$. A group of students enters the ...
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Alternative proof $g(u)=6 + 5 \sin u + \sin(2 u)- \cos u - \cos(2 u) \ge 0$ for $u\in\left[-\frac \pi 2, \frac \pi 2\right]$

The given inequality $$g(u)=6 + 5 \sin u + \sin(2 u)- \cos u - \cos(2 u) \ge 0$$ for $u\in\left[-\frac \pi 2, \frac \pi 2\right]$, comes out from an answer given to this other recent question. The ...
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Proving triangle midsegment theorem without quadrilaterals or similarity

I'm here with a question on a basic fact presented to you below: The midline of a triangle is parallel to the third side of that triangle and half its length. There is a known proof listed here on ...
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How can you prove directly that two polynomials with only real roots can make a polynomial that has complex roots with addition?

Generally, if a polynomial over the real numbers of degree $n$ has $n$ roots then it means that it only has real roots. I found an example where when I add two of these polynomial, I get one with a ...
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