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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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Determine where this function is continuous - floor function

Determine where this function is continuous: $$f(x)= \begin{cases} \frac{1}{x-\lfloor x\rfloor}, & \text{for } x \notin \mathbb{Z}\\ 1, & \text{for } x \in \mathbb{Z}\\ \end{cases} $$ ...
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3answers
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Proving that a line segment joining the midpoints of two sides of a triangle equals half of third side.

Is this proof correct? I know it can be proven using the Midpoint Theorem. I believe the proof is correct, in that case, please explain how I should prove it with the same approach without a graph. If ...
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4answers
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How many triplet primes of the form $p, p+2, p+4$ are there? Prove your conjecture.

I am giving this problem to 8th grade students, and I am hoping that people can help me find elementary ways to prove this problem. I would love to find other arguments that are accessible to 8th ...
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0answers
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Another proof for impossibility of covering $\mathbb{R}^{n}$ with a set of varieties of cardinality less than $2^{\aleph_0}$

Let $\mathbb{R}^{n}$ be the affine $n$-space. Let $\{V_{\alpha} \}_{\alpha \in \Gamma}$ be a set of real varieteis such that $\mathbb{R}^{n} \subseteq {\bigcup V_{\alpha}}.$ Prove that $|\Gamma| \geq ...
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Another proof for $\lim\limits_{n \to \infty}n^{\frac{1}{n}}=1(n=1,2,\cdots)$. [closed]

Problem Prove$$\lim\limits_{n \to \infty}n^{\frac{1}{n}}=1$$where $n=1,2,\cdots$ Proof Notice that $$1=1^{\frac{1}{n}}\leq n^{\frac{1}{n}}=\sqrt[n]{1\cdot 1\cdot 1\cdots\sqrt{n}\cdot \sqrt{n}}\leq \...
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1answer
86 views

Find condition for $\alpha + \beta+ \gamma=\alpha . \beta. \gamma$

Find condition for real numbers $\alpha , \beta, \gamma$ such that $\alpha + \beta+ \gamma=\alpha . \beta. \gamma$ Solution: we have this set for example: $\sqrt 3+\sqrt 3+\sqrt 3=\sqrt 3.\sqrt 3.\...
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1answer
93 views

The number of permutations of $\mathbb{N}$

Let $\mathbb{N}!$ be the set of permutations of the natural numbers. With $\aleph_0 = \text{card}\ \mathbb{N}$ you need two facts of cardinal arithmetic to show that the number of permutations of $ \...
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1answer
21 views

A second approach to a proof of the External Angle Theorem for triangles

Hello, I have been given the task shown to solve. I have done so. I am now trying to find a second way of solving thus being able to anticipate any answer someone may have when solving. I was able ...
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1answer
17 views

Intuition about combining half-open intervals into inequality with absolute value?

I'm think about how to use the two conditions $$0\le r\lt n\\ 0\le r'\lt n,$$ to prove $\lvert r'-r\rvert\lt n,$ and the way I achieved this is by backwardly expand the result into $$-n\lt(r'-r)\lt ...
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2answers
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Prove that 3,5,7 are only 3 consecutive odd primes. [duplicate]

I have done a number theory proof. Showing that if a prime $p > 5$ then atleast one of $p-2,p+2$ is a multiple of 3. And when $p=3$ then that multiple is 3 itself. However this question was ...
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1answer
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Groups are cocomplete by adjoint functor theorem without explicit use of free products

$$ \newcommand{GRP}{\mathsf{GRP}} \newcommand{I}{\mathcal{I}} \newcommand{im}{\mathrm{Im} \;} $$ This is a follow up for the question Royal Road to Free Groups and Free Products. I was exploiring ...
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1answer
52 views

An elementary proof of $\pi_1(S^n) =1$ using this general theorem

I was looking for an elementary proof of $ S^n$ being simply connected. My favourite is the one given by the user Olivier Bégassat in this post. Then i realized that what he did is a particular case ...
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3answers
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Prove that if $f(x)=0$ except for a finite number of points, then $f\in\mathcal{R}[a,b]$ and that $\int^{b}_{a}f(x)dx=0.$ [closed]

Suppose $f:[a,b]\to\Bbb{R}$ and that $f(x)=0$ except for a finite number of points $c_1,c_2,\cdots,c_n$ in $[a,b]$. Prove that $f\in\mathcal{R}[a,b]$ and that $\int^{b}_{a}f(x)dx=0.$ My pdf presents ...
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Inversion in a sphere preserves circles (proof)

Inversion in the unit sphere, for a vector $x$, is defined by $$\frac1x = \frac x{x^2} = \Big(\frac1{x\cdot x}\Big)x$$ How can we prove that a circle's inversion in the sphere is also a circle? (I ...
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2answers
108 views

Dynamical systems proofs wanted

Recently I read dynamical systems proofs of the irrationality of $\sqrt{2}$ and Fermat's theorem. Now, I'm interested in other dynamical systems proofs of well-known things similar to this. What book, ...
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2answers
37 views

Alternative proof for subbasis for order topology and product topology: Finite intersections of elements of $\mathscr S$ is a basis

Munkres Topology: Both of the following proofs are at the very end of, respectively, Sections 14 and 15, which seem to correspond with the definition of a subbasis' being at the very end of Section 13....
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2answers
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Prove by induction that $\sum_{n=1}^\infty \frac{1}{2^n} = 1$

The title explains the problem fairly well; is there a way to prove by induction that $\sum_{n=1}^\infty \frac{1}{2^n} = 1$. If not are there other ways? I have thought of showing it by rewriting ...
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0answers
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Does this prove the Inscribed Rectangle/Square Theorem?

Imagine a circle, with four points on it forming a rectangle. Twisting and bending this circle along those four points, any curve can be produced. $\blacksquare$ Is this a proof of the Inscribed ...
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Calculus Circle Related rates circle proof

NOTE: THIS QUESTION HAS ALREADY BEEN ANSWERED AND CAN BE FOUND AT THIS LINK (Related rates circle problem) Question: Two circles $A$ and $B$ have the same center. The radius of the inner circle $A$ ...
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2answers
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proof that $[0,1]$ is compact

I tried coming up with a proof of compactness of $[0,1]$ in $\mathbb{R}$ and thought of the following method. please let me know if it is correct or how it could be made more correct. For any open ...
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1answer
35 views

Alternate way to show these two integrals are the same

Let $T>0$ and $a,b,x\in \mathbb{R}$ such that $a<b$. Take $I:=\int_{-T}^T \frac{e^{-it(x-b)} - e^{-it(x-a)}}{-it} dt$. Take $J:=2 \int_{x-b}^{x-a}\frac{\sin(Tu)}{u}du$. One way to show $I=J$ ...
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1answer
25 views

“Every convergent sequence is bounded” and the choice of epsilon

In the proof of the theorem which states that every convergent sequence is bounded, it's chosen $\varepsilon=1>0$, but the proof works for every $\varepsilon>0$ right? Couldn't the proof be ...
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0answers
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Is there a combinatorial proof of this recurrence relation for families of sets thus defined?

Define $S_{k,n}$ like in this question. I think I have proved the identity $$\sum_{m\in S_{k,n+1}}m^{-1}=\sum_{m\in S_{k,n}}m^{-1}+\frac1{p_{n+1}}\sum_{m\in S_{k-1,n}}m^{-1} $$for all $n\ge k\ge1$. In ...
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0answers
26 views

What are elementary proofs in the context of geometry and arithmetic?

In his answer to the question Is an integer a sum of two rational squares iff it is a sum of two integer squares? André Nicolas mentions that the proof that if $m$ is a positive integer, then $m$ ...
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1answer
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Congruence equation for distinct primes $p,q$.

Prove that, for distinct primes $p,q$: $$\tag{1} p^{q-1} + q^{p-1} \equiv{1} \pmod{pq}.$$ This is a pretty straight forward problem, I know, and although the standard proof is simpler, I was ...
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Geometrical proof of the impossibility of angle trisection by straightedge and compass

There's a fascinating (almost) geometrical proof of the impossibility of angle trisection by straightedge and compass by Terrence Tao which is somehow more comprehensible (and more directly to the ...
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Proof of Laplace Transform of *Real Number* Powers

As I understand it, the theorem and proof for the Laplace transform of positive integer powers is as follows: Theorem Let $t^n : \mathbb{R} \to \mathbb{R}$ be $t$ to the $n$th power for some $...
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1answer
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Alternative proof for chains

Let $(X,\le)$ be a chain, $S\subseteq X$, and suppose $\sigma=\sup S$ exists. Then for every $x\in X$ with $x<\sigma$, there is some $s\in S$ s.t. $x<s\le\sigma$. By contradiction: Let $x\in X$ ...
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3answers
145 views

How show that $\sum_{j=0}^\infty \frac{(-1)^j\binom{\frac12}{j}}{{2j+1}}=\frac{\pi}4$

As discussed in detail in that OP Find the limit of $\lim _{ n\rightarrow \infty }{ \sum _{ k=1 }^{ n }{ \frac { \sqrt { { n }^{ 2 } - { k }^{ 2 } } } { { n }^{ 2 } } } } $, using Riemann sum we ...
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6answers
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What is the relation between 'the order of $x^k = n/{\gcd(k,n)}$' and Lagrange's Theorem?

Algebra by Michael Artin Cor 2.8.11 to Lagrange's Theorem (Theorem 2.8.9). Question: What is the relation between Cor 2.8.11 and the order of $x^k$ given by $n/{\gcd(k,n)}$ (in the text, ...
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2answers
83 views

Proof verification $(x_1^2 + x_2^2)(y_1^2 + y_2^2) = (x_1y_1 + x_2y_2)^2+ (x_1y_2 - x_2y_1)^2$ (Spivak's Calculus)

I have to prove the Schwarz inequality with that, then I want to know if what I did was so rigorous or not (And if it's good). $$(x_1^2 + x_2^2)(y_1^2 + y_2^2) = (x_1y_1 + x_2y_2)^2 + (x_1y_2 - ...
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1answer
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How would Euclid have proved that $ a \times b = ((ab)/c) \times c$?

The following statement seems too obvious to prove or even to mention: That the area of a rectangle with side lengths $a$ and $b$ equals the area of a rectangle with side lengths $(ab)/c$ and $c$ for ...
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1answer
38 views

Determining whether proofs of the Pythagorean theorem are essentially the same, or essentially different

There are great annotated lists of proofs of the Pythagorean theorem, to name just two: cut-the-knot.org brilliant.org There's an obvious difference between geometric and algebraic proofs (which ...
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Alternative proof of Levy's continuity theorem(problem with understanding specific part)

I found alternative proof of Levy's continuity theorem. In spite of reading this proof 12 times and being able to find explanation to therse parts, which were not clear for me, there is 1 crucial ...
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2answers
28 views

Proof of a practical method of a natural number's equation

Example Question : $3m + 4n = 70$, $m,n$ are natural numbers. How many values can $m$ have? I learned a method to solve this kind of problem, but I've never thought about that before. for n=1 =...
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5answers
112 views

Prove that $1+x<e^x$ for all real numbers $x\neq0$.

Prove that $1+x<e^x$ for all real numbers $x\neq0$. Here is my solution: We'll prove this as two cases; first when $x<0$ and then when $x>0$. Let $\alpha>0$. Then, $$e^\alpha-\alpha e^\...
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1answer
96 views

Prove cyclic subgroup has n distinct elements $\langle x \rangle = \{1,x,x^2,..,x^{n-1}\}$ without using Euclidean algorithm again

Algebra by Michael Artin Prop 2.4.2 Here is the statement and the proof: I split Prop 2.4.2(c) into Parts I, II and III respectively: Part I: There is some positive integer $n$ s.t. $S=\mathbb ...
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1answer
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Questions on the length of a positive integer $a\in \mathbb{Z}^+$.

Definition. We call length of a positive integer $a\in \mathbb{Z}^+$, and we write $\ell(a)$, the number of $2$-adic digits of $a$. That is $\ell(a):=\lfloor \log _2 a\rfloor +1$. I found on my ...
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2answers
51 views

Proof about a theorem that says a function is continuous if only if f is right continuous in a and left continuous.

Theorem: A function $f:D\to \mathbb R$ is continuous in $a\in D$ $\iff f$ is left and right continuous in $a$. Proof: I firstly thought just to write down the definitions of left and right ...
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0answers
75 views

Riemann surface of $f(z)=\sqrt{z^2+1}$

Construct the Riemann surface of $f(z)=\sqrt{z^2+1}$. Notice that $f(z)=\sqrt{z^2+1}=\sqrt{z+i}\sqrt{z-i}$. Define $f_1(z)=\sqrt{r_1r_2}e^{\frac{\theta_1+\theta_2}{2}}$ where $r_1=\vert z-i\vert,r_2=...
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1answer
80 views

Proof explanation: Prove that $\Vert f(b)-f(a)-f'(a)(b-a) \Vert\leq \sup_{x\in [a,b]} \Vert f''(x)\Vert\Vert b-a\Vert^2$ [duplicate]

Two days ago, I asked a question Prove that $\Vert f(b)-f(a)-f'(a)(b-a) \Vert\leq \sup_{x\in [a,b]} \Vert f''(x)\Vert\Vert b-a\Vert^2$ but was answered just once. However, I am finding it ...
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1answer
40 views

Convergence of a logarithmic sequence

Is the sequence $\left\{\ln\left((1+\frac1n)^n\right)\right\}_{n=1}^{\infty}$ convergent or divergent?. I tried to solve it by L Hospital's rule and arrived at 0...implying it is convergent..is it? ...
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0answers
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Trying to prove BIBO stability implies the boundedness of the impulse response of an LTI system without a counterexample

I have tried to prove that BIBO stability implies that the impulse response of the LTI system, can someone please check if the proof is correct?Proof part 1/2 proof part2/2
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0answers
41 views

Riemann Integrability of a function $f$ on $[0,1]$

The question goes like this: Suppose is a bounded function such that $f$ is Riemann integrable on $[a,1]$ for every $a \in (0,1)$. Is $f$ Riemann integrale on $[0,1]$? Duplicate My approach: ...
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6answers
276 views

Can we prove that $\lim_{n\to\infty} \frac{1^{n_0}+2^{n_0}+\cdots+n^{n_0}}{n^{n_0+1}}$ is finite for any $n_0\in\Bbb N$ without a direct computation?

Can we prove without direct calculation that this limit is finite for any natural number $n_0 \in \mathbb{N}$? $$ \lim_{n \to \infty} \frac{1^{n_0}+2^{n_0}+\cdots+n^{n_0}}{n^{n_0+1}} $$
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1answer
55 views

Application of Implicit Function Theorem: Problem 2-40 from Spivak's Calculus on Manifolds

At the end of the section on Implicit Functions in Spivak's Calculus on Manifolds, we have Problem 2-40: Problem 2-40. Use the implicit function theorem to re-do Problem 2-15(c). And, Problem 2-15(...
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0answers
32 views

Is there any proof of cauchy formula for repeated integration without induction?

https://en.wikipedia.org/wiki/Cauchy_formula_for_repeated_integration Here, the formula has been proved by using mathematical induction. Is there any other way to prove it? I mean how did cauchy ...
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0answers
45 views

Let $a,b\in\mathbb{R}^* = [-\infty,\infty]$ be such that $a\leq b + \varepsilon$ for all $\varepsilon>0$. Show $a\leq b$.

I wish to show the above statement. I've shown this to be true by taking cases of $a$ and $b$. For $a,b\in\mathbb{R}$, I assumed $a>b$ and derived a contradiction. The cases involving $\pm \...
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2answers
37 views

Differentiation of solution to time-dependent system of equations: Problem 2-15(c) from Spivak's Calculus on Manifolds

This is Problem 2-15(c) from Spivak's Calculus on Manifolds. Problem 2-15(c). If $\det(a_{ij}(t)) \neq 0$ for all $t$ and $b_1,\dots,b_n : \mathbb{R} \to \mathbb{R}$ are differentiable, let $s_1,\...
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1answer
56 views

Matrices of rank 1 and the inequality $rank(A+B) \leq rank(A) + rank(B)$

I'm interested in a proof of the inequality $rank(A+B) \leq rank(A) + rank(B)$ that does not use the bases of $A$'s and $B$'s row/column spaces, as described in other answers on this site. My idea is ...