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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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Show that a linear map on a finite dimensional complex vector space always have an eigenvalue.

What is an alternative proof that a linear map $T$ on a finite dimensional complex vector space $V$ with dimension $n$ always has an eigenvalue? Here is the original proof idea: We take a no zero ...
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0answers
20 views

Convexity of Sum-of-max-linear-terms

Consider the sum-of-max-terms function $f: \mathbb{R}^m \mapsto \mathbb{R}$: $$\begin{align} f(z) = \sum_{k=1}^K \underset{j \in \mathcal{I_k}}{\max} \{z_j \} \end{align}$$ where $\mathcal{I}_{k} \...
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5answers
31 views

what's the best way to prove the equivalences of such formulas?

I want to prove the following: $$2^n+2^{n-1}+...+2^1 + 1 = 2^{n+1}-1$$ The only Method that I know of is proof-by-induction but is this the best way to prove the equivalences of such formulas?
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7answers
70 views

Showing $a^2 + b^2 > 2ab$ without using the fact that $(a-b)^2 = a^2 + b^2 -2ab$?

I am wondering if we can Show that $a^2 + b^2 > 2ab$ without using the fact that $(a-b)^2 = a^2 + b^2 -2ab$? (I'm particularly interested in $0<a<b<1$ but I don't think restricting $a$ ...
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3answers
76 views

Proving a function $\frac{1}{2y}\int_{x-y}^{x+y} f(t) dt=f(x)$ is a linear polynomial

Here's the question: Let $f:\mathbb{R}\to\mathbb{R}$ be a twice differentiable function such that $$\frac{1}{2y}\int_{x-y}^{x+y} f(t) dt=f(x)\text{ for all }x\in\mathbb{R},y>0$$ Show that ...
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1answer
43 views

Proving the divergence of $\sum_2^\infty{\frac{1}{n \log n}}$ using comparison test

It is quite straightforward using the Cauchy Condensation test. But is there any way to solve this problem using some well known comparison test? I cannot think of any way of my own. Any help/hint ...
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3answers
63 views

Given $f$ is continuous and $f(x)=f(e^{t}x)$ for all $x\in\mathbb{R}$ and $t\ge0$, show that $f$ is constant function

This question was asked in ISI BStat / BMath 2018 entrance exam: Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function such that for all $x\in\mathbb{R}$ and $t\ge 0$, $$f(x)=f(e^{t}x)$$ ...
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1answer
33 views

Proving a sequence is bounded under given conditions

This question was asked in ISI BStat / BMath entrance exam: Let $f :\mathbb{R}\to\mathbb{R}$ be a differentiable function such that its derivative $f'$ is a continuous function. Moreover, assume ...
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2answers
87 views

Prove recursive formula for number of spanning trees in complete graph.

When $t(n)$ is number of spanning trees in complete graph $K_n$ prove recursive formula for $t(n)$: $$t(n) = {1\over (n-1)}\sum_{k=1}^{n-1} k(n-k){n-1 \choose k-1}t(k)t(n-k)$$ Could someone prove it ...
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2answers
168 views

Quartic as a product of quadratics

The statement is Every degree $4$ polynomial with real coefficients is expressible as the product of two degree $2$ polynomials with real coefficients. This and much more general versions are of ...
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1answer
53 views

Proving if $\sum_{n=1}^{\infty} a_n$ and $\sum_{n=1}^{\infty} b_n$ both converge then the series $a_1+b_1+a_2+b_2+\ldots$ converges.

If $\sum_{n=1}^{\infty} a_n$ and $\sum_{n=1}^{\infty} b_n$ both converge with sums $\alpha$ and $\beta$, then show that the series $a_1+b_1+a_2+b_2+a_3+b_3+\ldots$ converges with sum $\alpha + \...
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1answer
24 views

How can I show these series combinations are the same algebraically?

We have to show: $$\frac{1}{2}\sum_{r=1}^{n} {\frac{1}{r}} \ + \ \sum_{r=1}^{n} {\frac{1}{2r+1}} \ = \ \sum_{s=1}^{2n}{\frac{1}{s+1}}$$ I understand how this works intuitively and the most formal ...
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1answer
68 views

Sketches of proofs written in English that show that first-order logic is complete?

I came across the following in a textbook: $\mathcal{L}$ is a language. $\alpha \in \mathcal{L}$ is a formula. $\alpha \frac{c}{x}$ means: replace every instance of $c$ in $\alpha$ by $x$. I am ...
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2answers
53 views

A question regarding the Bolzano Weierstrass Theorem

This is probably a very stupid/absurd question.... The theorem states that every bounded real sequence has a convergent subsequence. We further know that every sequence of real numbers has a monotone ...
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0answers
39 views

How do I prove this inequality of a convex function on an interval

Suppose h is a continuous function on the interval $[0:1]$ such that for any $t_1, t_2 \in [0,1]$ $$ h\left(\frac{t_1 + t_2}{2}\right) \leq \frac{h(t_1) + h(t_2)}{2}$$ Show that for all natural ...
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1answer
57 views

Proof by induction involving symmetric groups

I am aware that i have posted this question before, but the comments i received did not really help silly old me. So i would appreciate it if someone can walk me through some parts of the problem. I ...
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3answers
28 views

About cluster points in a proof

I am reading in a proof, but there's only one part that I do not understand Theorem: Let $\Omega$ be an open subset of $\mathbb{C}$, let $z_1,z_2,\dots$ be a sequence of different points from $\...
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1answer
235 views

Is induction neccessary for proving that every injective mapping of a finite set into itself is a mapping onto itself?

Upon reviewing the basic theorem that the number of elements in a fixed finite set is unique, I tried to determine what part of this proposition is in need of proof. It seems axiomatic. Nonetheless, ...
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0answers
79 views

If $f$ is continuous on $[a,b]$ and $g$ equals $f$ except at $x=c\in(a,b)$ then $\int_{a}^{b} f = \int_{a}^{b} g$.

If $f$ is continuous on $[a,b]$ and $g$ equals $f$ except at $x=c \in (a,b)$ where $g$ is defined arbitrarily. Then prove that $g$ is integrable on $[a,b]$ and $\int_{a}^{b} f = \int_{a}^{b} g$. Here'...
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1answer
116 views

Evaluating $I(x)=\int_{0}^{\frac{\pi}{2}}{\frac{du}{x^2\cos^2u+\sin^2u}}$

The following is from a national junior contest of an african country. Find the integral $$I(x)=\int_{0}^{\frac{\pi}{2}}{\frac{du}{x^2\cos^2u+\sin^2u}}$$ $$\underline{\textbf{My attempt:}}$$ ...
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2answers
76 views

Calculate $\sum_{n=1}^{\infty}\left(\frac{1}{3}\frac{2}{5}\cdots\frac{n}{2n+1}\frac{1}{n+1}\right)$

Denote $a_n=\frac{1}{3}\cdot\frac{2}{5}\cdots\frac{n}{2n+1}\cdot\frac{1}{n+1}$. Please prove $$\sum_{n=1}^{\infty}a_n=\frac{\pi^2}{8}-1$$ This is the answer given by my friend. He also used ...
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1answer
63 views

Proof verification: Squeeze theorem for integrals

This was a problem in my analysis textbook- Suppose that $f(x)\le g(x)\le h(x)$ for all $x\in[a,b]$ and that $\int_{a}^{b}f$ and $\int_{a}^{b}h$ exists and are equal. Prove that $\int_{a}^{b}g$ ...
1
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1answer
29 views

Proof Verification, family of sets

Given $\left\{ A_i \right\}_{i\in\mathbb{N}}$ a family of non empty sets, $A$ a non empty set such that $\forall i\, \in \mathbb{N}, A_i\cap A\neq\emptyset$. The family $\left\{B_j \right\}_{j\in\...
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1answer
58 views

Find $P(n+1)$ for a polynomial P

Let $P(x)$ a n-degree polynomial such that for all $k\in\{0,1,...n\}$, we have $P(k)=\frac{k}{k+1}$. Find $P(n+1)$. My attempt: I consider a new polynomial $Q(x)=(x+1)P(x)-x$. $Q$ is a n+1-degree ...
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3answers
34 views

Show if all continuous functions $f$ from $X \to \{0,1\}$ with the discrete topology are constant, then $X$ is connected.

This problem was similar to this, but I couldn't make it work with the direction I want to show: I tried showing the contrapositive (the inverse of the linked problem), which is showing "if $X$ is ...
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2answers
67 views

Where does poisson distribution functional form come from?

I know that the punctual probability function of a random variable $X$ with a Poisson distribution is: $$P(X=k)= e^{-\lambda }\frac{\lambda ^{k}}{k!}.$$ Also, I've learned that the formula can be ...
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2answers
68 views

Show that there exists a set $U$ which is both open and closed and $x \in U \subseteq V$.

Let $X$ be a compact topological space. Suppose that for any $x, y \in X$ with $x \neq y$, there exist open sets $U_x$ and $U_y$ containing $x$ and $y$, respectively, such that $$ U_x \cup U_y = X\...
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1answer
107 views

What is the correct way to solve the equation: $x^4-x^3+x^2-x+1=0$

Given the equation: $x^4-x^3+x^2-x+1=0$ we need to find both its real and complex roots. What is the easiest and correct method for solving the equation? Here is my approach, but it gives wrong ...
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0answers
29 views

Differentiability and continuity while partials have different conditions

The relevant things I read and will discuss are in this snapshot (from Folland Advanced calculus, and Wolfram Alpha, and this answer by zhw for an old question). Also, let me add two other links ...
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3answers
379 views

Analytic proof of area probability

$X,Y$ are i.i.d. $unif(-1,1)$ random variables. Prove that $$P(X^2+Y^2\leq 1)=\frac{π}{4}$$ Geometrically, I understand how that happens. $(X,Y)$ is a random point in square having centre at origin ...
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2answers
102 views

Fundamental theorem of algebra avoiding complex analysis

Suppose you have to give a proof of fundamental theorem of algebra to someone who did not know complex analysis. You can’t therefore use Liouville’s Theorem. How would you proceed avoiding complex ...
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0answers
36 views

What is a method to prove this trigonometric recursion without mathematical induction?

tl:dr at the end: So I started of scribbling this simple function in my notebook $ f(\theta) = (1+\sin\theta + \cos\theta)$ $ = \sin^2\theta + \cos^2\theta + \sin\theta + \cos\theta$ $ = (\sin\...
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0answers
39 views

What are alternative ways to show this infinite sum is the absolute error between $\pi$ and 22/7

tl:dr at the end. In my first year undergraduate course having definite integrals, I studied about the $\beta$ and $\Gamma$ functions. There was a problem from this particular exam which I never had a ...
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0answers
52 views

Is there better method than this one to solve quadratic diophantine equations?

Say we have the quadratic Diophantine equation : $4x^2+9y^2=100$ I'm not really sure what the usual method to find integer solutions to these equations are , but one way I thought of was the ...
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votes
3answers
53 views

Showing $\langle x,y\mid x^2, y^3, xyxy^{-1}, (xy)^7\rangle$ is trivial.

I encountered this problem in Sims' "Computation with Finitely Presented Groups". Show that $\langle x,y\mid x^2, y^3, xyxy^{-1}, (xy)^7\rangle$ is trivial. The book uses coset enumeration or ...
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1answer
42 views

How to prove this simple property for two sets?

We are given two vectors, $a = (a_1,\dots,a_n)$ and $b = (b_1,\dots,b_n)$ such that $0 \le a_i=b_i \le \varDelta$ for $i=1,\dots,n$. We want to modify each of these vectors in an iterative procedure ...
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1answer
105 views

Prove the inequality $\frac{1}{1+x^2}+\frac{1}{1+y^2}+\frac{1}{1+xy} \geq \frac{3}{1+\frac{(x+y)^2}{4}}$ when $x^2+y^2=1$

I have to prove the inequality $$ \frac{1}{1+x^2}+\frac{1}{1+y^2}+\frac{1}{1+xy} \geq \frac{3}{1+\frac{(x+y)^2}{4}} $$ when $x^2+y^2=1$, using Cauchy-Schwarz Inequality. The RHS is equal to $\frac{12}...
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0answers
24 views

If $G$ is a graph having $p$ vertices and min degree $\delta(G)\ge\frac{p-1}{2}$, then the edge connectivity is $\delta(G)$.

Theorem: If $G$ is a graph having $p$ vertices and $\delta(G)\ge\frac{p-1}{2}$, then the edge connectivity of $G$ is $\delta(G)$. $\delta\left(G\right)$ denotes the minimum degree of $G$. ...
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1answer
27 views

number of ways to order 5 english and 4 french hits

the question is as follows: suppose a dj has 5 english hits and 4 french hits. in how many ways can these songs be played if no two english hits should follow each other? in how many ways can these ...
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3answers
32 views

Inequality involving absolute values.

I want to ask is whether there is a method to solve following inequality more easily and compactly or it is the only method. $$|x-2|+|x-8|\le x-2$$ What I know is taking $x<2,8>x>2,x>...
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0answers
71 views

Baby Rudin 6.11 and 6.8

Theorem 6.8 If $f$ is continuous on $[a, b]$ then $f \ ; \epsilon \; R(\alpha)$$f$ is Riemann integrable on$ [a, b]$. $f \epsilon R(\alpha)$ means $f$ is Riemann integrable w.r.t. $\alpha$ ...
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1answer
33 views

Show that if $A$ is star convex, $A$ is simply connected.

It is answered a couple times as I can see but I want to know if my proof is working. I think I didn't understand the definition of trivial fundamental group very well. Here it goes: A is simply ...
1
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1answer
55 views

Is there a geometric proof of $\frac1r = \frac{1}{h_a} + \frac{1}{h_b} + \frac{1}{h_c}$ in a given triangle?

Can anyone provide a geometric proof of $\frac1r = \frac{1}{h_a} + \frac{1}{h_b} + \frac{1}{h_c}$ in a given triangle, if there is one? Here, $h_i$ refers to the distance from the side $i$ to the ...
2
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1answer
42 views

Parallelogram Law Geometric Proof

So, I've dealt with the parallelogram law in various ways a bunch of times now. And algebraically (or maybe I mean arithmetically) it makes perfect sense to me--I can prove it, understand it, and I ...
15
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6answers
672 views

Very indeterminate form: $\lim_{x \to \infty} \left(\sqrt{x^2+2x+3} -\sqrt{x^2+3}\right)^x \longrightarrow (\infty-\infty)^{\infty}$

Here is problem: $$\lim_{x \to \infty} \left(\sqrt{x^2+2x+3} -\sqrt{x^2+3}\right)^x$$ The solution I presented in the picture below was made by a Mathematics Teacher I tried to solve this Limit ...
2
votes
1answer
92 views

Taking $\int_{-\infty}^{\infty} \frac{e^{i \omega x}}{1 + ix}dx$ through Residue Theory?

In the text "Basic Complex Analysis" Third Edition by Jerrold E. Marsden and Michael J. Hoffman I'm inquiring if there's an alternate way through Complex-Analysis to evaluate $\text{Example (4.38)}$ ? ...
1
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0answers
23 views

Understanding logical implication and alternate proofs

I just don't get exactly how logical implications work. When using truth tables (albeit impractical for large amounts of variables), it can be quite simple to show truths equate for something such as ...
4
votes
2answers
62 views

Find the last three digits of $p$ if the equations $x^6 + px^3 + q = 0$ and $x^2 + 5x - 10^{2013} = 0$ have common roots.

Find the last three digits of $p$ if the equations $x^6 + px^3 + q = 0$ and $x^2 + 5x - 10^{2013} = 0$ have common roots. Let $a,b $ be the solutions of second equation, then by Vieta we have $a+...
5
votes
1answer
62 views

sum of the first $n^2$ natural numbers closed form

Before I get downvoted I am still a beginner so please bare with me. I know the summation of the first n are $\frac{n(n+1)}{2}$. Does that imply the sum of the first $n^2$ is $\frac{n^2(n^2+1)}{2}$?
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2answers
52 views

Derivations of the trapezoid rule

I know the general method to derive the trapezoid rule is with Taylor series, or, you know, to just look at the trapezoids and figure out the rule. However, I feel that for such a simple rule, there ...