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

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58 views

Linear maps preserving the determinant and Hermiticity

Conjecture: Let $H_n$ be the space of $n\times n$ complex Hermitian matrices and let $\varphi:H_n \to H_n$ be a linear map which preserves determinants: \begin{equation} \det \circ \varphi = \det. ...
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5answers
47 views

Let $\{a_n\}$ be a sequence, $a,b$ given so that $\lim\limits_{n\rightarrow\infty} a_n=a$ and $\lim\limits_{n\rightarrow\infty} a_n=b$, then $a=b$.

Let $\{a_n\}$ be a sequence. If $a,b$ given so that $\lim\limits_{n\rightarrow\infty} a_n=a$ and $\lim\limits_{n\rightarrow\infty} a_n=b$, then $a=b$. Proof: Let $\epsilon>0$. Since ...
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0answers
15 views

Is there other ways to show reverse triangle inequality

I know that for $x,y \in \mathbb{R}$ we have that $$|x-y| \ge ||x|-|y||$$ which can be proven by writing $$|x|=|x+y+(-y)|$$ and $$|y|=|y+x+(-x)|$$ and applying triangle inequality. But I am ...
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1answer
83 views

Proof that $\lim_{x\to0}\frac{\sin x}x=1$

Is there any way to prove that $$\lim_{x\to0}\frac{\sin x}x=1$$ only multiplying both numerator and denominator by some expression? I know how to find this limit using derivatives, L'Hopital's rule, ...
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18 views

Is the following proof correct of why $\gcd(a,b)$ smallest linear combination of $a$ and $b$?

This is the proof I have: Lets see why $\gcd(a, b) $ is the smallest positive linear combination of $a$ and $b$: Let $LC = \{ s'a + t'b : s', t' \in \mathbb{Z}, s'a + t'b > 0 \}$. By the ...
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2answers
63 views

Is this a correct proof of why $\gcd(a,b) = \gcd(b, a- b)$?

I have a proof but I wasn't sure if it was correct (or how rigorous it is). I will point out what worries me. Let $D_a = \{ d : d \mid a\}$ (i.e. all elements that divide $a$) and similarly $D_b = \{ ...
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40 views

How do you show if $d \mid a$ and $d \mid b$ then $d \mid \gcd(a,b)$ without knowing that $\gcd(a,b)$ is a linear combination of $a$ and $b$?

I was trying to prove that if $d \mid a$ and $d \mid b$ then $d \mid \gcd(a,b)$ but wanted a proof that didn't require me to know that $\gcd(a,b) = ax + by$, i.e. that didn't require me to know that ...
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2answers
62 views

Is there a way to show that $\gcd(a,b) = ax + by $ without also showing that its the smallest positive linear combination?

Is there a way to show that $\gcd(a,b) = ax + by$ without also showing that it is the smallest positive linear combination? i.e. Can it be shown that there exists an $a$ and $b$ such that $\gcd(a,b) = ...
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128 views

How to show this cover of $\mathbb{Q}$ doesn't cover $\mathbb{R}$?

Let $\{q_n : n \in \mathbb{N}\}$ be an enumeration of $\mathbb{Q}$ and define $\mathcal{O} = \{I_n : n \in \mathbb{N}\}$ being $$I_n = \left(q_n - \frac{1}{2^n}, q_n + \frac{1}{2^n}\right).$$ It is ...
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5answers
49 views

Proof of sets. Need an example

My question is to show that $X-(Y \cup Z)$ is a subset of $(X-Y) \cup (X-Z)$. I already did the proof for that and understand that but the second part is to give an example to show that in general, ...
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1answer
94 views

Starting index of a sequence is irrelevant

"Let $(a_n)_{n=m}^{\infty}$ be a sequence of real numbers, let $c$ be a real number, and let $k \geq 0$ be a non-negative integer. Show that $(a_n)_{n=m}^{\infty}$ converges to $c$ iff ...
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1answer
28 views

Solution Sets of Homogeneous Systems

I had to prove the following theorem: Suppose that $A\mathbf x=\mathbf b$ is consistent for some given $\mathbf b$, and let $\mathbf p$ be a solution. Then the solution set of $A\mathbf x=\mathbf b$ ...
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5answers
121 views

A combinatorial proof of $\forall n\in\mathbb{N},\,\binom{n}{2}=\frac{n(n-1)}{2}$

The property $\forall n\in\mathbb N,\,\binom{n}{2}=\frac{n(n-1)}{2}$ was given in our first chapter on probability theory among binomial coefficients' properties. It is really easy to prove with the ...
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3answers
138 views

Elementary theorems that require AC

It seems that AC is hiding (maybe concealed?) even in some elementary results. An example: Theorem: Let $X \subseteq \mathbb R$ and let $x_0 \in \mathbb R$ be an accumulation point of $X$. Then ...
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1answer
83 views

$T:X\to Y$ a linear operator, $Y$ finite-dimensional, then $\ker T$ is closed iff $T$ is continuous

I have to prove that if $T:X\to Y$ a linear operator on normed spaces, $Y$ finite-dimensional, then $\ker T$ is closed iff $T$ is continuous. On a lot of places I see a proof that looks like this: it ...
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1answer
36 views

Prove that $x\cdot y=0 \iff x=0$ or $y=0$ without cases

I have proved that given $x,y \in F$, $F$ a field, $x\cdot y=0 \iff x=0$ or $y=0$ by making cases for neither $x$ nor $y$ equals $0$ (and did a proof by contradiction) and then two cases for $x=0$ and ...
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1answer
32 views

Synthetic proof of a bisector length

In a exam the following was asked and I found a proof based on a law of cosines. Is there a pure synthetic proof without trigonometry? $ABC$ is a triangle with side lengths $CA=a,CB=b$. $D\in AB$ is ...
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30 views

Show that $[H,K,L] = A_5$, where $H,K,L$ have order $2$.

Let $H, K$ and $L$ be subgroups of order $2$ in some group $G$, and observe that the set $\{[h, k, l] : h \in H, k \in K, l \in L\}$ contains at most one nonidentity element, and so it generates a ...
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1answer
30 views

Showing that a wave equation $u_{tt}=ku_{xx}$ has solution $u=0$ if $u(x,0)=u_t(x,0)=0$

Show that a wave equation $\rho u_{tt}=Tu_{xx}$ has solution $u=0$ if $u(x,0)=u_t(x,0)=0$. Thoughts: This is easy using the general solution to wave equations ...
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2answers
35 views

Show that $[a,b] \subset Range(f)$

Question. Show that $[-\frac{1}{2},\frac{1}{2}]\subset Range(f)$, where $f(x)=\frac{x}{x^2+1}$. My proof. Let $y=\frac{x}{x^2+1}\Leftrightarrow yx^2-x+y=0.$ For real values of y, values of x must ...
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1answer
73 views

Contractions of maximal ideals in finite type $K$-algebras

In my commutative algebra class we proved the following theorem: Contractions of maximal ideals: Let $A\xrightarrow{\ \ f\ \ }B$ be a homomorphism of $K$-algebras. Suppose that $B$ is of finite ...
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13 views

Other proofs of uniqueness of interpolating polynomial

I think that one of the well known proofs is this one: Let $f:[a,b]\to\mathbb{R}$ be a function and $P_n:[a,b]\to\mathbb{R}$ be the interpolating polynomial for $f$ on $[a,b]$. Let the nodes of ...
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1answer
70 views

The value of $\gcd(2^n-1, 2^m+1)$ for $m < n$

I've seen this fact stated (or alluded to) in various places, but never proved: Let $n$ be a positive integer, let $m \in \{1,2,...,n-1\}$. Then $$\gcd(2^n-1, 2^m+1) = \begin{cases} 1 ...
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3answers
38 views

Proof of $\lim_{k\to \infty}{|x^k-x|\over { 1+|x^k-x|}}=0$

Is there an easy way to prove that $$\lim_{k\to \infty}{|x^k-x|\over { 1+|x^k-x|}}=0$$ with $x\in \mathbb R$ without using $\epsilon-\delta$ definition? Any help would be really appreciated
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1answer
33 views

Different ways of Proving the existence of Tensor Product

This is Just a curosity. Let $A$ be a commutative ring and $M,N,P$ be A-modules.I know that tensor product of $M$ and $N$ is a universal object ($ M \otimes N$,u) (where $M \otimes N$ is a $A$-module ...
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0answers
66 views

(ZF - Foundation) proves that $\log_2$ can't be infinitely iterated: Alternative proof

I think I solved Ex. (12) in Chapter I of Kunen's book. It states that ZF sans Foundation proves: For every set $X$, $$ \aleph(X) < \aleph(\mathcal{P}^3(X)), $$ where $\aleph(X):= \sup\{\alpha \in ...
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1answer
132 views

Alternative proof of the Riemann Sum Theorem using Mean Value Theorem for Integrals.

I've been reviewing proofs for a couple of calculus theorems and as I was trying to recall the proof of the Riemann Sum Theorem which uses Lower Sums and Upper Sums I came up with an idea to prove it ...
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1answer
25 views

Show that if there is a function $g:Y\to X$ such that $g\circ f$ is the identity function on $X$, then $f$ is one to one.

Let $X,Y$ be nonempty sets and $f:X\to Y$ be a function. Show that if there is a function $g:Y\to X$ such that $g\circ f$ is the identity function on $X$, then $f$ is one to one. My approach is by ...
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2answers
26 views

Clarification: Prove there exists a number $N$ such that $n > N$ implies $s_n >a$

Below is the proof that I have been working on and the solution provided by the professor. Let $(s_n)$ be a convergent sequence, and suppose $\lim s_n > a$. Prove there exists a number $N$ ...
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1answer
23 views

Find a criterion for divisibility

Find a criterion such that $\displaystyle\sum_{i=1}^ni$ divides $\displaystyle\prod_{i=1}^ni^2$ for $n\in\mathbb N$. What I have done so far, $\displaystyle\sum_{i=1}^ni=\frac{n(n+1)}{2}$ and ...
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1answer
67 views

Proving that and how $ \frac{1}{n}\sum\limits_{p\le n}\lfloor n/p \rfloor - \sum\limits_{p\le n} 1/p $ approaches $0$

Let $p$ denote a generic prime number. By Mertens' second theorem, the sequence $$\sum\limits_{\ p \le n} \frac1p - \log\log n$$ converges to the Meissel-Mertens constant $M\approx 0.2614972$. Now let ...
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1answer
46 views

Cantor's diagonal argument modified version

I have the following doubt regarding Cantor's diagonal argument. First of all, the "usual case" is quite clear for me. If $X$ is some set, then we can show there is no surjection from $X$ onto the set ...
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2answers
60 views

Prove that no set can contain everything (or every other set)

Prove that there cannot exist a set that contains everything. Ill put my proof in the answer so please check it there. Also if there is a more creative way to do this(using the basic axioms) if it's ...
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1answer
36 views

An alternative method to find $\sum_{k=1}^{2n-1} | \beta ^k - 1|$

Let $\beta \in \mathbb{C}$ such that $\beta ^n = 1$ but $ \beta ^k \neq 1$, $\forall k=1,2,\cdots, n-1$. Find the value of $$ \sum_{k=1}^{2n-1} | \beta ^k - 1|.$$ I came across such a question ...
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1answer
66 views

How to prove triangle inequality in How to Prove It Sec. 3.5 Question 12c?

(a) Prove that for all real numbers $a$ and $b$, $$|a| \le b \text{ iff } -b \le a \le b.$$ (b) Prove that for any real number $x$, $$-|x| \le x \le |x|.$$ (Hint: Use part (a).) (c) Prove that ...
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3answers
104 views

Proving uniqueness of solutions to $\sin^2A + \sin^2B = \sin (A+B)$ without using multivariable calculus

In the course of solving a trigonometric problem (see $a^2+b^2=2Rc$,where $R$ is the circumradius of the triangle.Then prove that $ABC$ is a right triangle), in one approach the following equation ...
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2answers
72 views

$\mathbf{Set} \not \simeq \mathbf{Set}^*$ by considering $\{1, 2 \} \to \{1\}$

This answer gives a nice way of seeing why the category of sets is not isomorphic to its dual. I would like to know whether there is a proof from a certain different direction. When considering the ...
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1answer
51 views

Proof that $A\cap\emptyset=\emptyset$

I'm trying to prove $A\cap\emptyset=\emptyset$. I've seen several proofs for this which all seemed to essentially go about proving it by noticing that $\emptyset\subset A\cap\emptyset$ by definition ...
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1answer
51 views

About a proof regarding a property of groups of order $pq$ where $p$ and $q$ are primes

I'm studying right now Automorphisms in Dummit & Foote's Abstract Algebra (Section 4.4). In pages 135-136, the following example is given: and here's Proposition 16 muntionned in the ...
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82 views

Showing that only $(n+1)^{n-1}$ of all the possible $n^n$ choices assure a full car park

This exercise is taken from the site of Queen Mary University of London: A car park has $n$ spaces, numbered from $1$ to $n$, arranged in a row. $n$ drivers each independently choose a favourite ...
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39 views

Uniform Continuity of $\frac {1}{x}$ on [$a, \infty$) for positive $a$

$\frac {1}{x}$ behaves nicely in that it's monotone and the derivative is monotone also. So on [$a,\infty$] it can be seen that the $\delta$ which will work everywhere is the $\delta_1$ at the end ...
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1answer
100 views

Is this alternative proof of Theorem 3.7 (“Baby” Rudin, Ch. 3) correct and, if so, well written?

Rudin, in his Principles of Mathematical Analysis, proves the following theorem: The subsequential limits of a sequence $\{p_n\}$ in a metric space $X$ form a closed subset of $X$. I've tried to ...
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5answers
293 views

How to prove $3^\pi>\pi^3$ using algebra or geometry?

It's a question of a some time ago test, I've found a way to solve the problem using calculus, but always I've thought that exist a solution with algebra and geometry. Thank you for your time.
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57 views

Proof that 10 lines pass through the centroid of a triangle

Let $A$, $B$, $C$, $D$, and $E$ be points on a circle. For any three points, we draw the line going through the centroid of the triangle formed by these three points that is perpendicular to the line ...
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2answers
48 views

Verifying a Proof for Spivak's Calculus Question (Chapter 2 Problem 9)

It says "Prove that if a set $A$ of natural numbers contains $n_0$ and contains $k+1$ whenever it contains $k$, then A contains all natural numbers $\ge n_0$". Am I allowed to construct another set ...
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2answers
109 views

prove if f(x) has an infinite limit then limit of 1/f(x) is = 0

I wanted to ask if someone can do me the favor pointing out the mistakes I might of made in proving the theorem below. Also is there a way to prove the theorem without using the definition of limits? ...
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2answers
41 views

Confusion with the reconstruction conjecture?

After reading about the reconstruction conjecture for graphs, I came up with what I thought was a proof by contradiction. Consider the class $T$ of (isomorphism classes of) finite graphs, and the ...
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3answers
73 views

How to prove a specific sequence is Fibonacci's with no prior knowledge nor trial and error?

Let $n$ be a positive integer and let $s_n$ be the number of increasing sequences of integers, alternatingly even and odd, starting with $0$ and ending with $n$. E.g. for $n=3$ we only have the two ...
2
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3answers
117 views

Another identity with binomial coefficients

I'm looking for an easy way to prove this identity $$\sum_{j=0}^{n}{(-1)^j j (n-j) {n \choose j}} = 0$$ for $n > 2$. I know this can be proven by differentiating $(1+x)^n = \sum x^j{n \choose j}$ ...
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32 views

Rational analogue of expansion to base b

As is well known, we can expand every positive integer $n$ to a base $b \in \Bbb N$ in the form $$n = \sum_i a_ib^i ,\ \ \ 0\leq a_i \leq b_i-1$$ uniquely. Less well known is that we can do this for ...