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

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7 views

Prove that the number of nodes in ORBDD for fn with given order On is 2n +2

I cannot embed image yet, so I have no choice but to include a link here. (Also if anyone can include a link/tutorial/guide to how to display notations, I'd really appreciate it) This image contains ...
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2answers
20 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 ...
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1answer
19 views

Multidimensional Cantor diagonal argument for ordering infinite sets [duplicate]

Cantor diagonal argument is a powerful proof technique. It has been used for a lot of proofs. For instance, it has been used to prove that $|\mathbb{N}| < |\mathbb{R}|$. What can we say about the ...
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3answers
108 views

A purely algebraic proof of $\vec{a}\cdot \vec{b} = \lVert\vec{a} \rVert\lVert\vec{b} \rVert\cos(\theta)$

I have seen a proof of the fact that $$ \vec{a}\cdot \vec{b} = \lVert\vec{a} \rVert\lVert\vec{b} \rVert\cos(\theta) $$ where $\vec{a}$ and $\vec{b}$ are two vectors. The proof relies on the Law of ...
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0answers
17 views

How can I verify the following equality?

$$\int_0^{\infty}\frac{C\exp(-\frac{mx^2}{\Omega})}{\Omega^m}\frac{1}{\sqrt{2\pi}\lambda\Omega}\exp\left(-\frac{(\ln ...
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1answer
49 views

The well-ordering principle implies Zorn's Lemma

I have read and understood proofs for each implication between $AC$, $ZL$, $WO$ except this one. These proofs need about 10 lines each. Can someone share a neat, hopefully short, proof for $WO\implies ...
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15 views

How does one get $p=2$ from a condition that there be non-trivial linear transformations of every dimension that to any power are $p$-norm-preserving?

Verifying that (p=2) satisfies $$\forall n\in\mathbb{Z}^+.\exists A\in(\mathbb{R}^{n\times n}\setminus\{I_n\}).\forall k\in\mathbb{R}.\forall ...
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62 views

Unramified at a point $x \in X$ if and only if $\Omega _{X,x} = 0$

This is Corollary 6.2.3 in Liu's book. Let $f: X \to S$ be a morphism of finite type of locally Noetherian schemes. Then $f$ is unramified at a point $x \in X$ if and only if $\Omega_{X/S, x} = ...
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3answers
60 views

Alternative Proof: if $n$ is an integer, prove that $\frac{n ( n^4 - 1)}{5}$ is an integer

I have proven this by the induction method but would like to know if it can be proven using an alternative method.
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1answer
26 views

Probability space for zebras and their number of stripes

On a trip to Africa the researcher Alison notices that zebras with an even amount of stripes have double the probability to be seen than zebras with an odd amount of stripes. Let $E_n$ denote the ...
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1answer
34 views

Show that finite dimensional subspace is closed

We know that if $V$ is a normed vector space and $W$ is a finite dimensional subspace of $V$, then $W$ is closed. One way to prove this is to show that $W$ is actually complete. Since complete space ...
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1answer
17 views

Prove directly from definition: countably compact subsets of metric spaces are closed

I am trying to prove the statement that every countably compact subset Y of a metric space (X,d) is closed. I am aware of the fact that, for metric spaces, countable compactness is equivalent to ...
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3answers
23 views

How to determine intervals where $f$ is greater than $g$?

I have two functions, $f(x) = 2x$ and $g(x) = \frac{x^3}{3}$. I solved for $x$ where $f = g$, finding $x = \pm 6^{1/2}$, then solved for $x$ where $f > g$, $x > \pm 6^{1/2}$, and where $f < ...
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1answer
33 views

Reference request: alternative proof for every open set in $\mathbb{R}^n$ can be expressed as countable disjoint union of open boxes

A "box" is a cartesian product of intervals of the type $[a,b]$ I am using Terence Tao's introduction to measure theory and on page 24 a proof of title statement is given, however, it is quite ...
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2answers
24 views

Show that $\bigcup_{n=1}^\infty A_n= B_1 \backslash \bigcap_{n=1}^\infty B_n$

Let $\{B_n\}$ be a decreasing set $B_1 \supseteq B_2 \supseteq B_3 \supseteq ....$ Define $A_n = B_1 \backslash B_n$ i.e. $A_1 = \varnothing, A_2 = B_1 \backslash B_2$ If we imagine $\{B_n\}$ as a ...
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3answers
66 views

Prove that the function $f:\mathbb{R}^2\rightarrow\mathbb{R}$ is constant

Suppose that the function $f:\mathbb{R}^2\to\mathbb{R}$ has first order partial derivatives and that $$\frac{\partial f}{\partial x}(x,y)=\frac{\partial f}{\partial y}(x,y)=0\qquad\text{for all ...
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3answers
119 views

How do you prove this without using induction?

How do you prove this without using induction $$\frac{1}{n}+\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{2n-1}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots+\frac{1}{2n-1}$$
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1answer
274 views

Is there an easy way to see that this simple recurrence is 9-periodic?

In a colloquium talk yesterday, Robert Bryant pointed out that for all initial values $a_0, a_1 \in \mathbb{R}$, the sequence generated by the recurrence relation $$ a_{n+1} = |a_n| - a_{n-1} $$ turns ...
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1answer
32 views

Geometric Proof of DeMoivre's Formula

I want to know if there were a geometric proof of DeMoivre's formula. My attempt was starting with an easy complex number and observing patterns, then generalizing that pattern. If you start with ...
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29 views

More elegant derivation of the shift in median bin occupancy

In answering Median of a multinomial variable, I found to my own surprise through a somewhat tedious calculation that the expected value of the median of the ball counts in $3$ bins into which $n$ ...
2
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2answers
193 views

Calculus approach to solve this Quadratic equation problem

Both roots of the equation $$(x-b) (x-c) +(x-a) (x-c) +(x-a) (x-b) = 0$$ are always positive , negative or real. Prove your result. By solving this equation I got $3x^2 - 2(a+b+c)x +ab + bc + ca ...
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3answers
39 views

Intuitive proof for a Combinatorial Problem

Given a set $S$ such that $|S|=N$ and $S$ contains exactly $K$ $0$s $(K >0)$ and $N-K$ $1$s, then exactly half of the subsets of $S$ contain an $odd$ number of 1s, $indepedent$ of the value of ...
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1answer
14 views

A short proof that the set of discontinuities of a regulated function is countable

Does a short proof exist for the following theorem? The set of discontinuities of a regulated function on $[a, b]$ is countable.
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1answer
27 views

Proving sequence converges

I am trying to prove: Suppose that {an} and {bn} are two sequences such that {an} and {an + bn} converge. Prove that {bn} converges. Here is my first attempt: Proof: Suppose that {$a_n$} and {$b_n$} ...
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3answers
52 views

Prove if $f$ is continuous at a and $g$ is discontinuous at a, then $f + g$ is discontinuous at a

Hello I want to prove: if $f$ is continuous at $a$ and $g$ is discontinuous at $a$, then $f + g$ is discontinuous at $a$. But with the $\epsilon - \delta$ definition of continuity and discontinuity (I ...
3
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0answers
73 views

Most complicated proof of Pythagoras

Usually a mathematician aims for clarity and elegance when conducting a proof. However, the antimathematician buries all hope of assimilating intuition and reasoning. To illustrate this, I seek the ...
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1answer
10 views

Exact differential equation $f(t,x)+g(t,x)\dot x=0$ and constants

The differential equation $$f(t,x)+g(t,x)\dot x=0\tag{*}$$ with $(t,x)\in U\subset\mathbb R^2$ and $U$ open is called exact, if there is a continuous and differentiable function $F\colon ...
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2answers
48 views

(Correct Proof?) Show that every convergent sequence $(\mathbf{x}_{k})$ in $\mathbb{R}^{n}$ is bounded.

Now, this is a pretty simple proof and I just wanted some more experienced members here to have a look at it and maybe give me feedback on my proof idea for the statement in the title. I also found ...
8
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1answer
55 views

Evaluating a certain integral without the fundamental theorem

I'm a TA for a calculus course. And they recently began calculating definite integrals using a definition equivalent to Riemann's criterion. Of course, the type of things they were calculating were ...
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2answers
53 views

Find $\int_{-1}^{1} \frac{\sqrt{4-x^2}}{3+x}dx$

I came across the integral $$\int_{-1}^{1} \frac{\sqrt{4-x^2}}{3+x}dx$$ in a calculus textbook. At this point in the book, only u-substitutions were covered, which brings me to think that there is a ...
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1answer
64 views

Prove that the determinant of an invertible matrix $A$ is equal to $±1$ when all of the entries of $A$ and $A^{−1}$ are integers.

Prove that the determinant of an invertible matrix $A$ is equal to $±1$ when all of the entries of $A$ and $A^{−1}$ are integers. I can explain the answer but would like help translating it into ...
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5answers
126 views

Finding $\int \frac{dx}{a+b \cos x}$ without Weierstrass substitution.

I saw somewhere on Math Stack that there was a way of finding integrals in the form $$\int \frac{dx}{a+b \cos x}$$ without using Weierstrass substitution, which is the usual technique. When $a,b=1$ ...
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1answer
58 views

Can I assume that random variables with exponential distribution are positive?

Let $(Y_n)$ be i.i.d random variables following exponential distribution with parameter $1$. Let $X_n=\min(Y_1,\dotsc, Y_n)$ Prove that $ X_n \xrightarrow{P} 0$ It's easy to prove that ...
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2answers
41 views

Number of graph vertices of odd degree is even

This elementary result is normally stated as a corollary to the Handshaking Lemma, which says nothing about it other than that it's true. I wonder if there is more depth to this fact, in particular if ...
3
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0answers
68 views

Every skew-symmetric matrix has even rank [duplicate]

Let $F$ be a field where $char(F)\neq2$ and let $A$ be a skew-symmetric matrix over $F$. Prove that rank of $A$ is even. I think the best way to prove it, is using induction on size of $A$. for ...
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2answers
60 views

How to evaluate $\int \frac{dx}{(1-x)\sqrt{1-x^2}}$ without a trig substitution/parts?

I'd like to find $$\int \frac{dx}{(1-x)\sqrt{1-x^2}}$$ but without using a trig substitution or integration by parts. I can already see that $x=\sin \theta$ works it out quite nicely, but I was ...
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2answers
83 views

Find all $n$ such that $\sqrt{5n+2}$ is an integer.

Here is my solution. There is no such $n$. If $n$ is odd, then, then $5n+2 \equiv 7 \pmod {10}$. Else, $5n+2 \equiv 2\pmod {10}$. But, the quadratic residues of $10$ are only $0,1,4,9,6,5$. ...
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1answer
26 views

Proofing de Movire without Induction and in a neat way

The "usual way" gone for proving de Movire is via the road of induction. However this road get tiresome and thus wondered, if there were another way. However I came up with a proof that relies on ...
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1answer
27 views

Show that $\mathbb{Q}(\sqrt{2})$ is a field.

Proof: Since $\mathbb{Q}$ is a field, then $\mathbb{Q}$ is a domain. (Theorem: if $R$ is a domain, then $R[x]$ is a field.) By the theorem, $\mathbb{Q}[x]$ is a field. So, letting $x = \sqrt{2}$, ...
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2answers
56 views

On $\sigma-$ finite space $fg\in L^1$ for every $g\in L^q$ prove $f\in L^p$

Let $(X,\mathcal{A}, \mu)$ be an measure space. Let $f$ be an extended complex-valued $\mathcal{A}-$measurable function on $X$ such that $|f|<\infty$ $\mu$-a.e. on $X$. Suppose that $fg\in ...
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0answers
64 views

$fg\in L^1$ for every $g\in L^1$ prove $f\in L^{\infty}$

Let $(X,\mathcal{A}, \mu)$ be an arbitrary measure space. Let $f$ be an extended complex-valued $\mathcal{A}-$measurable function on $X$ such that $|f|<\infty$ $\mu$-a.e. on $X$. Suppose that ...
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0answers
27 views

Theoretical Math Sequence Proof

Suppose that {xn} is a sequences such that every subsequence {xni} has a subsequence {xnmi} that converges to x. Show that {xn} is bounded. I tried to do a proof by contradiction but am not sure if ...
2
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1answer
71 views

Prove that if $A$ and $B$ are square matrices and $AB$ is invertible, then both $A$ and $B$ are invertible

I already know how to prove this using the definition of inverse and the associative property of matrix multiplication, but I was wondering if this would also be a valid proof. As $A$ and$ B$ are $n ...
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1answer
39 views

Proving that a set has no largest member

Here's the question in mind. Let $$A = \left\{r : r \quad \text{is a rational number and} \quad r^2 < 2\right\}$$Prove that $A$ has no largest number. (Hint: if $r^2 < 2$, and $r > 0$, ...
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2answers
80 views

Proof that the difference of two positively squared integers never equals 1

This type of question is usually solved by a proof by contradiction, however I believe I have a direct proof of it, and I would like to know if its correct. Problem : Prove that there does not exist ...
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1answer
18 views

Prove that $|x-a_1|+|x-a_2|+|x-a_3|\geq a_3 - a_1$, for $a_1<a_2<a_3$, and determine the condition for equality.

I got this question from the first chapter of Courant and John's Introduction to Calculus and Analysis I. The problem is as follows: Prove that $|x-a_1|+|x-a_2|+|x-a_3|\geq a_3 - a_1$, for ...
1
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1answer
36 views

Primes Between $n$ and $2n$ For $n\ge6$

For $n\ge6$, there are at least two primes in the interval between $n$ and $2n$. Does anyone know of an already established and accepted proof for this? A reference would be helpful. I have read in ...
4
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0answers
83 views

Using the IVP definition of $\cos$ and $\sin$, how can we show that $\cos^2(x)+\sin^2(x) = 1$ without any “magic”?

One way to define the exponential, hyperbolic and circular functions is to assert that they're the unique solutions to certain IVP systems: The exponential function: $$\exp'(x) = \exp(x), \qquad ...
0
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1answer
121 views

Different proofs for $n ( n + 1 ) ( n + 2 ) ( n + 3 )$ [closed]

Different proofs that show $n ( n + 1 ) ( n + 2 ) ( n + 3 )$ cannot be the square of an integer, where n is a natural number.
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2answers
45 views

Alternate Proof to $f(e_G)=e_H$

Is this proof correct? Proof: Let for all $a$ and $e_G \in G$ we know that if $f$ is a homomorphism from ${(G,*)}$ to $(H,o)$ then, $f(a*e_G)=f(a)=f(a)$ o $f(e_G)$. Similairly ...