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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If $f''(x)+2f'(x)+3f(x)=0$, then $f$ is infinitely differentiable

I came across this problem: which of the following statements are true regarding differentiability. Is the following statement true? If $f$ is twice continuously differentiable in $(a,b)$ and if for all $x\in(a,b)$ , $$f''(x)+2f'(x)+3f(x)=0$$, then…
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How to prove surjectivity of $f:\{1,\ldots,n\}\to\Bbb Z_n,\,k\mapsto[k]$? Alternative proofs for $|\Bbb Z_n|=n$?

Im trying to prove that $\Bbb Z_n:=\Bbb Z/n\Bbb Z$ have cardinality $n$ just using properties of rings, this mean that Im trying to do it without the use of any multiplicative inverse $z^{-1}\in\Bbb Q$. My try: if $|\Bbb Z_n|=n$ then must exist a…
Masacroso
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Approximation of Lipschitz function in uniform norm

Let $F =\left\{ f\in C^1[a,b] : || f|| \leq c, ||f'||\leq k\right\}$ and $G = \left\{f\in C[a,b] : || f|| \leq c, f \text{ is globally Lipschitz with Lipschitz constant }\leq k \right\}$ where $a,b,c,k$ are fixed real constants, and the norm is the…
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Can we specifically prove that if $n\in\mathbb{Z^+}$ is composite and square-free then the ring of integers of $\mathbb{Q}(\sqrt{-n})$ is not a UFD?

Let $\mathcal{O}_K$ be the ring of integers of a field $K$. I have learnt about the Baker-Heegner-Stark theorem, which implies that if $K=\mathbb{Q}(\sqrt{-n})$ with $n\in\mathbb{Z}^+$ square-free, then $\mathcal{O}_K$ is a UFD if and only if $n$ is…
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Symmetric group $\mathfrak{S}_4$ does not have a normal subgroup of order 2.

Prove: The symmetric group $\mathfrak{S}_4$ does not have a normal subgroup of order 2. I found the following way of showing this, which does not make me happy. My questions are: Is this proof correct? And: Is there a nicer way of showing this, by…
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Proof regarding the multiplicative group of integers modulo $p^m$

I have the following conjecture: Let $p$ be a prime, and $g\in(\Bbb{Z}/p\Bbb{Z})^\times$. If $g$ generates $(\Bbb{Z}/p\Bbb{Z})^\times$, then $g$ also generates $(\Bbb{Z}/p^m\Bbb{Z})^\times$ for all $m\in\Bbb{N}$ with $m>1$. Now I did attempt a…
user3002473
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Compute $\sup\limits_{x\in (0,2)}\left|\operatorname{cosh}\left(x^2-x-1\right)\right|$

$$\rm{ Calculate: }\quad \displaystyle \sup_{x\in (0,2)}\left|\cosh\left(x^2-x-1\right)\right|$$ My proof: $$f(x)=\cosh\left(x^2-x-1\right)$$ $$f'(x)=\left(2x-1 \right)\sinh\left(x^2-x-1\right) $$ $$f'(x)=0 \iff x=\dfrac{1}{2} \rm{ or }…
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How to show that $h$ is continuous?

Claim. Let $f,g:X\to Y$ be two continuous functions where the topology on $Y$ is order topology. Define $h:X\to Y$ as $h(x)=\max\{f(x),g(x)\}$. Then show that $h$ is continuous. My Attempt (very rough sketch) Let $V$ be an open set in $Y$. Then…
user170039
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Prove that if f is diff. on $(-\infty,\infty)$ such that $f(0)=0$ and $f(1)=1$,then there is a $c\in(0,1)$ such that $f'(c)=2c$

I have been solving last exam questions of calculus course and encountered with a problem which I couldn't solve completely.The question is following; Determine whether the statement is true or false, and prove it; Suppose that f is differentiable…
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Prove "Two parallelograms on the same base and in the same parallels, are equal."

I understand Euclid's way of proving this. But, the book also says that I can prove this by decomposing one parallelogram into pieces, and then forming another parallelogram by combining those pieces together. I was thinking of dividing one…
user3000482
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If an invertible matrix $A$ commutes with $C$, show that $A^{-1}$ commutes with $C$

This is taken from a problem in Introduction to Linear Algebra by Strang: If an invertible matrix $A$ commutes with $C$, show that $A^{-1}$ commutes with $C$ This is what I tried to do, take note of the specific order in which I've multiplied the…
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Proof of Riemann Rearrangement Theorem

I'm reading the proof of Riemann Rearrangement Theorem in T. Tao's Analysis 1 textbook which can be found here Rearrangement Thm (the parts missing from the textbook, left as exercises for the reader, are completed by the user asking the…
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Check my proof on showing $|H|=|\mathcal{H}||K|$, where $|H|$ and $|\mathcal{H}|$ are corresponding subgroup in the correspondence theorem?

Let $\phi:G\to\mathcal{G}$ be a surjective homomorphism with kernel $K$. There is a bijective correspondence between subgroups of $\mathcal{G}$ (where we denote as $\mathcal{H}=\phi(H)$) and subgroups of $G$ that contain $K$ (where we denote as…
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Necessary and sufficient condition for the union of two intervals to be an interval

I am trying to write down a proof of the following fact: Let $I$ and $J$ be non-empty intervals of $\mathbb{R}$ such that $\inf I \leqslant \inf J$. The union $I \cup J$ is an interval if and only if one of the following conditions holds: $I \cap…
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Brute force way to show that $\rho(x,y) = \min\{1, d(x,y)\}$ is a metric

Following a hint in Short proof that $\rho^\prime(x,y) = \min\{1,\rho(x,y)\}$ is a metric I would like to use the brute force method to show that the standard bounded metric is a metric $$\rho(x,y) = \min\{1, d(x,y)\}$$ By brute force I mean to…
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