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

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42 views

Prove that the directrix is tangent to the circles that are drawn on a focal chord of a parabola as diameter.

Question: Prove that the directrix is tangent to the circles that are drawn on a focal chord of a parabola as diameter. Here is a picture; What I have attempted; Let the parabola ...
3
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1answer
36 views

Is there no proof of Dirichlet's results on quadratic residues without analysis?

Wikipedia states that all known proofs of Dirichlet's results $$ L(1) = -\frac{\pi}{\sqrt q}\sum_{n=1}^{q-1} \frac{n}{q} \left(\frac{n}{q}\right) \gt 0 $$ and $$ L(1) = \frac{\pi}{\left(2-\left(\...
3
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1answer
21 views

Seeking Additional Solutions for the Number of Network Links

The Problem Show that the number of possible links in a computer network of $n$ computers ($n \in Z \land n \geq 1$) is $\frac{n(n-1)}{2}$ in as many ways as you can. My Work Solution 1 Given $n$ ...
1
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1answer
32 views

Pentomino Tessellation Explanation

I need to explain why this pentomino tessellates in a mathematically coherent way. Here is the pentomino and the tessellation I have made. This pentomino can be translated to form a diagonal ...
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4answers
37 views

Any n x n matrix A can be written as a sum A = B + C where B is symmetric and C is skew-symmetric.

How can i proof the following statement: "Any n x n matrix A can be written as a sum A = B + C where B is symmetric and C is skew-symmetric." i tried to work out the properties of a matrix to be ...
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1answer
32 views

How can i proof that every high order derivative of $\frac{1}{1+x}$ is equal to$ (-1)^kk!$ at point $0$.

In order to calculate the Taylor-Maclaurin polynomial $\frac{1}{1+x}$ of order $n$ at point $0$, i used the identity: $$\sum_{i=0}^n x^i + \frac{x^{n+1}}{1+x} = \frac{1}{1+x}\qquad (I)$$ and i ...
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5answers
275 views

Why are an even number of flips required to get back to the original list?

Consider the list of numbers $[1, \cdots, n]$ for some positive integer $n$. Two distinct elements $i$ and $j$ of the list can be switched in a so-called flip. For example, let $f$ be a flip that ...
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1answer
28 views

How can we fill in some missing details in this proof?

Let $(X,d)$ and $(Y,\rho)$ be metric spaces and $X$ is compact. Suppose $f:X\to Y$ and $f_n:X\to Y$ are continuous functions such that for every $x\in X$, $\rho(f_n(x),f(x))$ decreases to $0$ as $n\...
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2answers
46 views

Could someone please check my proof that $(x_n) \text{ is Cauchy in } X\implies (f(x_n)) \text{ is Cauchy in } Y$

Let $(X,d)$ and $(Y,\rho)$ be metric spaces and $f:X\to Y$. Show that if $f:X\to Y$ is uniformly continuous, then $$(x_n) \text{ is Cauchy in } X\implies (f(x_n)) \text{ is Cauchy in } Y$$ My ...
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3answers
27 views

Proof by contradiction for: Prove that there do not exist positive integers $m$ and $n$ such that $m^2 - n^2 = 1$

I am kind of stuck on a practice problem relating to proof by contradiction that goes as follows: "Prove that there do not exist positive integers $m$ and $n$ such that $m^2 - n^2 = 1$" For the ...
6
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3answers
136 views

Proving $\sum\limits_{k=0}^n \sum\limits_{j=0}^{n-k} \frac{(k-1)^2}{k!} \frac{(-1)^j}{j!} =1$ without character theory

Let $n \geq 2$ be an integer. I would like to prove the following identity in an easy way: $$\sum\limits_{k=0}^n \left( \frac{(k-1)^2}{k!} \sum\limits_{j=0}^{n-k} \frac{(-1)^j}{j!} \right)=1$$ You ...
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5answers
468 views

Visualization of surface area of a sphere

I help mentor some really young, bright kids in mathematics. We were looking at geometric properties of various shapes, and one of the kids noted that the surface area of a sphere $S = 4\pi r^2$ ...
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1answer
40 views

Show that if $X$ is bounded above, then there exists $y \in a + X$ such that $y$ is an upper bound of $X$

"Let $X \subset \mathbb{R}$ be nonempty and $a > 0$. Define $$a + X = \{a+x: x \in X\}$$ Show that if $X$ is bounded above, then there exists $y \in a + X$ such that $y$ is an upper bound of $X$"...
0
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1answer
48 views

Alternate proof for Vieta's formula (formula for the summing the roots of a polynomial)

I just saw Vieta's formula for the first time, where it was stated that given some polynomial $$p(x)=a_nx^n+\cdots+a_0,$$ let $x_1,\ldots,x_n$ denote its roots. Then $$\sum_{i=1}^n x_i=-\frac{a_n}{...
1
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1answer
36 views

In transitive (non-trivial) group action, there must be at least one group element without fixed point

Let a finite group $G$ act transitively on a finite set $S$ with $|S| \geq 2$. The problem is to show that not every $g \in G$ can have a fixed point in this action. I proved this on my own, but I'm ...
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3answers
60 views

Prove $\forall n \in \mathbb{N}$ where $ n \neq 1 , n + \frac{1}{n} > 2$ using completing the square.

I have got this far; I am only unable to understand how to finish the proof. $n>0 \implies n + 1/n > 0 \implies n + 1/n + 2 - 2 > 0 \implies {\big(\sqrt{n}+\frac{1}{\sqrt{n}}\big)}^2 - 2 >...
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2answers
88 views

Why we square while doing the proof of √2 is irrational? [closed]

When we prove that $\sqrt 2$ is irrational by the method of contradiction, we assume $\sqrt 2$ is a rational number: $\sqrt 2 = a/b$ Squaring both sides, $2 = a^2/b^2$. Here is my question: is ...
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0answers
37 views

Proof of the Schwarz Lemma (simplified Cauchy/Clairaut Theorem)

My lecture notes states the following lemma (sometimes called Cauchy's Theorem or Clairaut's Theorem) without proof: Lemma. (Schwarz) Assume that $v_\xi$, $v_\eta$, and $v_{\xi\eta}$ exist and are ...
7
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3answers
141 views

Proving $\pi \gt e+\frac{1}{e} \gt \pi-\frac{1}{\pi} \gt e$

I created this problem for myself as a fun exercise. I want to prove the following statement: $$\pi \gt e+\dfrac{1}{e} \gt \pi-\dfrac{1}{\pi} \gt e$$ I found that the following upper/lower bounds ...
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0answers
22 views

Prove for the family of operators $\{S(t)\}_{t>0}$ that $S'(0)$ exists

Let's consider a real-valued function $V\in L^\infty(\mathbb{R})$ and a family of operators $\{S(t)\}_{t>0}$ defined on $L^2(\mathbb{R})$ as follows: $$\qquad \qquad\left(S(t)f \right)(x) = \frac{...
7
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1answer
64 views

Prove that there exists a sequence $(x_n)$ such that $\sum_n a_n x_n$ diverges

So, here's a nice little result that I deduced using the closed graph theorem from functional analysis, but I'm wondering if there's a more elementary approach: Fact: Let $(a_n)$ be a sequence ...
3
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0answers
51 views

Identity theorem for $2\pi\mathrm i$ periodic function

Let $f$ be entire as well as real-valued along the lines $\operatorname{Im}(z)=0$ and $\operatorname{Im}(z)=\pi$. Show that $f$ is $2\pi\mathrm i$ periodic under these circumstances, that is $f(z+2\pi\...
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0answers
48 views

$A = \sum_{n=0}^\infty a_n$ and $b_n \to B$ implies $\sum_{k=0}^n a_k b_{n-k} \to AB$

This question is motivated by the answer With $y_n$ a sequence of real numbers, prove that if $y_n=x_{n-1}+2x_{n}$ converges then $x_n$ also converges, where essentially the following fact is used: ...
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0answers
26 views

Alternative proof : Group algebra contains all irreducible G-modules.

It is well-known that the group algebra $F[G]$ is a direct sum of irreducible $G$-modules. The proof in my text book is as follows Write $F[G] = \oplus_{i=1}^n V_i$, where $V_i$ is a set of ...
0
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2answers
44 views

For all real numbers x and y there is a real number $z$ such that $x + z = y − z$.

To Prove: For all real numbers $x$ and $y$ there is a real number $z$ such that $x + z = y − z$. Proof: $x+z=y-z \Rightarrow y-x=2z$. Since $y$ and $x$ are real numbers, $2z$ is also real. Therefore ...
5
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3answers
174 views

How to integrate $\int_{-3}^3 (x^2-3)^{3} \,dx$ without expanding the polynomial?

How can I integrate: $$\int_{-3}^3 (x^2-3)^{3} \,dx,$$ neither expanding the polynomial nor using the relationship between integral and derivatives? I mean, there is a way to compute this integral ...
0
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1answer
34 views

Understanding the specific interchange of sum and integral

Why is it allowed to write this way, $$\int_{0}^{m} \sum_{k=1}^{\infty}\frac{1}{2\pi k}\frac{\sin(2\pi k t)}{t^{s+1}}\,\mathrm{d}t=\sum_{k=1}^{\infty}\frac{1}{2\pi k}\int_{0}^{m} \frac{\sin(2\pi k t)}{...
0
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0answers
22 views

Is it possible to prove the completeness of $\ell^2$-space without using Minkowski inequality?

The "standard" proof of completeness of $\ell^2-$space uses Minkowski inequality. More specifically, the proof presented in Introductory Functional Analysis with Applications with Applications by ...
3
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3answers
44 views

Prove: if $f:\mathbb{N} \rightarrow A, g:\mathbb{N} \rightarrow B$ are surjections, there exists a surjection $h:\mathbb{N} \rightarrow A \cup B$

I chose my own sets here for A and B as countably infinite pairwise disjoint subsets of $\mathbb{N}$. Can I do this with finite subsets and get an easier answer with the same result? Suppose $A = \...
4
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2answers
144 views

Proof of a statement about eigenvalues and eigenvectors.

How can i proof the following: Let $\mathbb L: V\rightarrow V $ be a linear mapping. Let $v_1,v_2,..,v_n$ non-zero eigenvectors with eigenvalues $c_1,c_2,..,c_n$ respectively, also let the ...
1
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2answers
77 views

Prove that a function defined on points in a plane is zero

Let $n\ge3$ be an integer, and $f:P\to\mathbb R$ be a function defined on any point in the plane $P$, with the property that for any regular n-gon $<A_1A_2A_3\cdots A_n>$, $$f(A_1)+f(A_2)+f(A_3)...
0
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1answer
17 views

How to proof that the set of all $X$ such that $X.A{\ge} c$ to some real number c is convex?

How can i proof the following statement: " Let $\mathrm A\in \mathbb R^{n}$ and $\mathrm c\in \mathbb R$, the set $\mathbb S$ of all elements belonging to $\mathbb R^{n}$ and satisfying the ...
2
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1answer
60 views

Show that $\int_0^\infty\frac{1}{1+x^n}\,\mathrm dx = \frac{\pi/n}{\sin(\pi/n)}$ for $\mathbb{N}\ni n\geq 2$

Show that $$\int_0^\infty\frac{1}{1+x^n}\,\mathrm dx = \frac{\pi/n}{\sin(\pi/n)}$$ for $\mathbb{N}\ni n\geq 2$. Let $S=\{r\mathrm e^{\mathrm i\varphi}\in\mathbb{C} \mid 0\leq r\leq R,0\leq \varphi\...
0
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1answer
83 views

Obscure proof that $+$ and $\times$ are continuous?

I am looking for proof of $+$ and $\times$ are continuous operations without using the standard definition of continuity (1. $\epsilon-\delta$, or 2. preimage of open sets or 3. sequential ...
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1answer
32 views

Slicker way to prove $\rho$ is a metric

Let $d$ be a metric on $X$, and define $\rho: X^{2} \to \mathbb{R}$ as $$\rho(x,y)=\frac{d(x,y)}{1+d(x,y)}$$ The difficulty is in checking the triangle inequality. So, I can prove this by writing $f(t)...
0
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1answer
22 views

Sequence of continuous function converge uniformly to continuous function

Theorem: Let $(f_n)$ be a sequence of continuous functions. If $(f_n)$ converge uniformly to $f$, then $f$ is continuous. I don't like the Rudin's approach, then I try to do my own way, but it seems ...
0
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0answers
27 views

Shortest proof for: if $f$ is continuous in an interval $A$ then $\{x,y\in A:f(x)=f(y)\land x\neq y\}$ is empty or uncountable [duplicate]

Prove that a continuous function on an interval $A$ have zero or uncountable points such that $f(x)=f(y)$ for $x\neq y\in A$. I did a proof that I will trace briefly: When $\{x,y\in A:f(x)=f(y)\land ...
3
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1answer
105 views

How to integrate $\int{1\over \sqrt{x^2-1}}\mathrm d x$ another technique without use trigonometry

How can I integrate $\int{1\over \sqrt{x^2-1}}\mathrm d x$ without using trygnometry? I mean using other methods like substituition, integration by parts (I tried these two but I think I am not seeing ...
2
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4answers
111 views

Prove $\ \sin(x) < x \ \ \ \forall x \in(0, 2\pi)$

Problem : Prove $\sin(x) < x \ \ \ \forall x \in(0, 2\pi)$ Now I have a possible solution for this, using limits and the first derivatives of $\sin(x)$ and $x$, but I don't feel it's a very ...
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5answers
688 views

Proving $\sum_{k=1}^n k\cdot k! = (n+1)!-1$ without using mathematical Induction. [duplicate]

Possible Duplicate: Summation of a factorial This equation is given: $$ 1\cdot1! + 2\cdot2! + 3\cdot3! + \ldots + n\cdot n! = (n+1)! - 1 $$ I've solved it using mathematical induction but I'...
1
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1answer
58 views

How to integrate $\frac{dx}{(x^2+k^2)^m}$, with $m$ positive integer.

How can I integrate: $$\int \frac{1}{(t^2+k^2)^m}\, dt$$ without trygonometric substituition? where $t= (x+(p/2))$ and $k= (1-(p^2/4))$ coming from an equation with complex roots: $x^2 + px + q.$ ...
4
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1answer
88 views

Alternate proof of the integral: $\int_0^1 x^x(1-x)^{2x}\,dx\neq3/8$

I am looking into the integral: $$I=\int_0^1 x^x(1-x)^{2x}\,dx\neq\frac{3}{8}$$ How might you prove this to be true? What's tough is that the integral $$3/8\lt I<0.37503$$ numerically. I managed ...
3
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1answer
59 views

Proof of Tychonoff's Theorem for an undergrad

In the midst of learning about compactness I come across Tychonoff's Theorem: Let $\{X_i : i \in \mathcal{A}\}$ be any collection of compact spaces. Then $\displaystyle\prod_{i \in \mathcal{A}}X_i$...
3
votes
1answer
68 views

Elegant proof of an elementary result in Linear Algebra

I've been reading Hoffman Kunze, and I came across this theorem (theorem $9$) which has a long and tedious proof. I've been wondering wether there could be a more elegant proof. Theorem 9. Let $e$ ...
16
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2answers
894 views

A simply-connected closed surface is a sphere

From the Classification Theorem for closed (i.e. compact and boundaryless) surfaces, it follows that $S^2$ is the only closed surface with trivial $\pi _1$. That's easy because the fundamental group ...
4
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0answers
56 views

Shortest proof for showing $\mathbb{Z}[\frac{1+\sqrt{-19}}{2}]$ is a PID.

I'm looking for an easy proof for that $\mathbb{Z}[\frac{1+\sqrt{-19}}{2}]$ is a PID. One proof I know is to show that the field norm is a Dedekind-Hasse norm, but this proof is quite dirty( that it ...
1
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2answers
25 views

Example to show that unconditional convergence does not imply absolute convergence in infinite-dimensional normed spaces.

I tried to show that unconditional convergence does not imply absolute convergence in infinite-dimensional normed spaces using a direct proof, but unfortunately I did not succeed. The definition of ...
0
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0answers
35 views

Proof by induction of Gronwall's inequality

I've an exercise which is the following: Gronwall’s Inequality Let $A > 0, B \geq 0$. Let $(\epsilon_j)_{j \in \mathbb{N}}$ be a sequence of real numbers with $$|\epsilon_{j+1}| \...
1
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2answers
59 views

Alternative proof of Fundamental Lemma of Variational Calculus?

I am confused by one of the proof in the Calculus of Variations by Gelfand and Fomin. On page 9, we have Lemma: If $\alpha(x)$ is continuous on $[a,b]$, and if $\int_a^b \alpha(x)h(x)=0$ for every ...
1
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0answers
14 views

Proving three asymptotic identities (Murray (1984)'s Exercise 1.1.4)

(Context: I'm self-studying Murray (1984). I learned (and have forgotten quite a lot of) real and complex analysis. I'm willing to relearn and to look up references.) Problem: if $f=O(g)$, show that ...