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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Does this proof need the axiom of choice?

I tried to prove this statement for a non-empty set $A$ $$A ~~\text{finite or countably infinite} ~\Leftrightarrow~ \exists\varphi : \mathbb{N} \rightarrow A ~~\text{surj}.$$ The $\Rightarrow$ is pretty straight forward and does not involve my…
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Exercise: prove the well-ordering principle using Zorn's lemma

Exercise (from T.Tao's Analysis 1 textbook): Let $X$ be a set, and let $\Omega$ be the space of all pairs $(Y,\leq)$, where $Y$ is a subset of $X$ and $\leq$ is a well-ordering of $Y$. If $(Y,\leq)$ and $(Y',\leq ')$ are elements of $\Omega$, we say…
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Exercise: prove statement logically equivalent to Axiom of Choice

Exercise (from T.Tao's Analysis 1 textbook): (Part I) Let $X$ be a set, and let $\Omega\subset 2^X$ be a collection of subsets of $X$. Assume that $\emptyset\notin\Omega$. Using Zorn's lemma, show that there is a subcollection $\Omega…
lorenzo
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Uniform Boundedness Principle for $L^p$

Let $E$ be a measurable set, $1\leq p<\infty$, and $q$ the conjugate of $p$ (i.e., $\frac{1}{p}+\frac{1}{q}=1$). Suppose $\{f_n\}$ is a sequence in $L^p(E)$ such that for each $g\in L^q(E)$, the sequence $\{\int_Eg\cdot f_n\}$ is bounded. Show that…
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In search of easier manipulation of an inequality to prove it

Let the zigzag function defined as $$zz(x):=\left|\lfloor x+1/2\rfloor-x\right|,\quad x\in\Bbb R$$ Then for $k\in\Bbb Z$ we have that $zz(k)=0$, and $zz(\Bbb R)=[0,1/2]$, and $zz$ is increasing in any interval of the kind $[k,k+1/2]$, and decreasing…
Masacroso
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Show that $[0,1)$ has no maximum, i.e. $\not \exists \max[0,1)$

Show that $[0,1)$ has no maximum, i.e. $\not \exists \max[0,1)$ My Attempted Proof Assume $\max[0,1)$ exists and put $\alpha = \max[0,1)$. Now $\alpha < 1$ else $\alpha \not \in [0,1)$. Put $\gamma= 1- \epsilon$ where $0 < \epsilon \leq 1$. Then…
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Wikipedia's proof that $f : \mathbb{R} \to \mathbb{R}, x \mapsto x^2$ is not uniformly continuous

Prove that $f : \mathbb{R} \to \mathbb{R}, x \mapsto x^2$ is not uniformly continuous Now I actually ask this question, as Wikipedia's proof of this seems wrong to me. Wikipedia's Proof: $f$ is uniformly continuous on $\mathbb{R}$ if $\forall…
Perturbative
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Are the functions with the map $\sum a_n b^{-n}\mapsto\sum a_n c^{-n}$ continuous?

Let $(a_n)$ a sequence such that $0\le a_n<\min\{b,c\}$ and $b,c> 1$. Then define the function $$f:A\to[0,1],\quad \sum_{n=1}^\infty a_n b^{-n}\mapsto\sum_{n=1}^\infty a_n c^{-n}$$ where $A$ is the subset of $[0,1]$ where the function is…
Masacroso
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Borsuks' conjecture in 2d alternative proofs

While searching through different materials(textbooks, internet websites, publications), I came across the script about Borsuk's conjecture (staffwww.fullcoll.edu/dclahane/ma/watsontalk.pdf). In this pdf. file(pages 25-30) very simple, but with…
user379168
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Is this a valid alternative proof of the sum law in limits?

So sum law of limits tell us $\lim_{n\to\infty} (a_n+b_n)=X + Y$ if $\lim_{n\to\infty} a_n = X$ and $\lim_{n\to\infty} b_n = Y$ Here is my attempt to prove it. Proof Let $\frac{\epsilon}{2}>0$, then $\exists N_a,N_b:$ (1) $|a_n - X| <…
Lemon
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Trying to prove Theorem $3.9$ from $I^K$ *Convergence* by Sleziak & Macaj

From $I^K$ Convergence by Sleziak & Macaj Theorem $3.9$ says The equivalence of (iii), (iv), (v) can be easily shown by the standard methods from the measure theory. I know very little Measure Theory. So can I prove them with other methods? I've…
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How to show these sets are equal

Given a metric space $(X,d)$ with $A\subseteq X $, show that $\overline{X \setminus A} = X \setminus A^\circ$ I know that I have to show that $\overline{X \setminus A} \subseteq X \setminus A^\circ$ and that $X \setminus A^\circ \subseteq …
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Find all functions $f: \mathbb Z \to \mathbb Z$ such that $f(x+y)=f(x)+f(y)$

Let $x=0\,\, f(x+y)=f(0+y)$ every $y \in \mathbb{Z}$ So we get $f(0)+f(y)=f(y)$ then $f(0)=0$ $0=f(0)=f(x(-x))=f(x)+f(-x)=0$ Let $f(1)=k$ for some $k \in \mathbb{Z}$ $f(n)=f(1+1+1+...+1) (n \text{ many } 1)$           $=f(1)+f(1)+...+f(1) (n \text{…
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Suppose $Ax = 0\Longrightarrow Bx=0$, then $A=PB$?

Suppose $A$ and $B$ are not zero matrices. Suppose they have the same dimensions $n\times n$ and have real entries. Suppose that if $Ax = 0$ then $Bx = 0$. Is it true that there exists $C$ such that $A = CB$? Is is true that there exists $D$ such…
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Do we need the axiom of choice to prove L'Hopital's rule?

I was looking over an honors-calculus proof of L'Hopital's rule today, and I couldn't help but feel a sense of unease. The proof states that $f(x)$ and $g(x)$ are continuous on $[a,b]$; differentiable on $(a,b)$; $\lim_{x\to a^+}…
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