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Questions tagged [proof-writing]

For questions about the formulation of a proof. This tag should not be the only tag for a question and should not be used to ask for a proof of a statement.

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1k views

Cauchy-Formula for Repeated Lebesgue-Integration

Recently, I came across the following statements. They were annotated as consequences of Fubini's Theorem but neither proof nor reference were given. Let $f:[a,b]\times [a,b]\to\mathbb{R}$ be ...
9
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231 views

Olympic number theory problem: is this solution fine and sufficiently well written?

Determine all positive integers $m$ such that the ratios $$ \frac{2(5^m+5)}{3^m+1}\quad\text{and}\quad \frac{9^m+1}{5^m+5}$$ are both integers. Attempt at a solution: If the ratios are both ...
8
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0answers
169 views

Degree of hypersurfaces in Grassmannians

In the book Discriminants, Resultants, and Multidimensional Determinants of Andrei Zelevinsky and Izrail' Moiseevič Gel'fand, the authors give the following definition of degree of a hypersurface in a ...
8
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460 views

Is there a Sokoban level with such conditions

First of all, let me explain what Sokoban is. It is a logic game created in Japan and it literally means "warehouse keeper". It is a type of transport puzzle, in which the player pushes boxes or ...
7
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499 views

Proof Strategy for a Dynamical System of Points on the Plane

I have a rather simple-looking system which exhibits a particular behaviour in simulation, and I would now like to attempt to prove this formally. The problem is, I don't really know where to start, ...
6
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74 views

Show that $\forall n, a_n$ is an integer that is multiple of $n$.

Consider the sequence $\{a_n\}$ defined by $\;a_1=1,\; a_2=2,\; a_3=24\;$ and $$a_n=\frac{6a_{n-1}^2a_{n-3}-8a_{n-1}a_{n-2}^2}{a_{n-2}a_{n-3}},\; \forall n\geq 4.$$ Show that $\;\forall n,\; a_n\;$ ...
6
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342 views

Proving not equicontinuity in $\Bbb R$ but equicontinuity in any other closed subset of $\Bbb R$

Let $F = {f_{n} | n ∈\Bbb N }$ be an infinite collection of functions $f_{n}(x)=e^{−n(x−n)^2} , x ∈ \Bbb R$. Prove that $F$ is not equicontinuous on $\Bbb R$ but equicontinuous on $[−a, a]$ for any $a ...
5
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64 views

Generalized Taylor derivatives test

I am seeking for a proof of the generalized derivative test to find inflection points, minima and maxima. I am seeking for a proof that I read some time ago but can't find anymore. The thesis was that ...
5
votes
0answers
774 views

Is there such a thing as “finite” induction?

I am not sure of the terminology that I am looking for, but I would like to use an inductive proof on the following type of structure. I have something of the form, for every $n \geq 2$ and for any $1 ...
5
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172 views

Proof of inequality involving binomial coefficients

Proving things is not my forte. Stumbled upon following identity: $$\binom{n+k+1}{k}>\sum_{i=1}^k \binom{\alpha_i}{i}$$ For $\alpha_i<\alpha_j $ for $i<j$ $0\leq \alpha_i \leq n+k$ Also $...
5
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0answers
242 views

Geometric proof for Sophomore's dream

Is there a "visual proof" for sophomore's dream? $$\int_0^1 x^{-x}\,dx = \sum_{n=1}^\infty n^{-n}.$$ In the wikipedia article there are two algebraic proofs, but the integral and the summation has ...
5
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107 views

Proving $f$ cannot be onto

If $f$ maps finite sets $A$ to $B$ and $n(A) < n(B)$, prove that $f$ cannot be onto. Proof by contradiction: If $f: A→B$ and $n(A) < n(B)$, $f$ is onto. Since, by definition of a function, $a∈...
5
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0answers
270 views

$\zeta(2)=\frac{\pi^2}{6}$ proof improvement.

Recently in one of my calculus exercise I have made out a (quite novel to me) proof for $\zeta(2)=\frac{\pi^2}{6}$ via the famous infinite product below: $$\sin(x)=x\prod_{i=1}^{\infty}(1-\frac{x^2}{i^...
5
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0answers
2k views

The Lebesgue Criterion for Riemann Integrability — a proof without using the concept of oscillation.

I am trying to prove the Lebesgue Criterion for Riemann Integrability without using the concept of oscillation. The Lebesgue Criterion for Riemann Integrability states that if $ f: [a,b] \to \mathbb{...
4
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49 views

Prove that $X\times Y=\emptyset$ $\iff$ $X=\emptyset$ or $Y=\emptyset$.

Prove that $X\times Y=\emptyset$ $\iff$ $X=\emptyset$ or $Y=\emptyset$. My proof. I will do contrapositive. LHS. Assume $X\neq\emptyset$ and $Y\neq\emptyset$, so there is $a\in X$ and $b\in Y$ such ...
4
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104 views

Improving clarity and argumentation with hard-to-describe combinatorial proof

I'm doing undergraduate research and the content of my paper depends on the following lemma. I tried something like a combinatorial proof, but it is clearly not rigorous, partly because my argument is ...
4
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0answers
94 views

Prove that $||x+y|| \leq ||x|| +||y||$

From Munkres' Topology, I get this question. A hint suggests us to use a result from a previous subquestion. But it seems that I don't need to use the previous result to prove this. Can someone help ...
4
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0answers
223 views

Proof that a factorization domain is a unique factorization domain if and only if every irreducible element is prime.

Today in algebra class my professor proved, among other things, that a factorization domain is a unique factorization domain if and only if every irreducible element is prime. I had a hard time ...
4
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110 views

Have there been any (interesting) computer-aided proofs that weren't proof-by-exhaustion?

It seems to me like many of the most famous "computer proofs" were done by basically brute-forcing through all of the cases, such as the four color map theorem. Are there any good examples of computer ...
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124 views

How do you prove this implication using its contrapositive?

$∀x∈R,|2x-1|≤5$ and $|2x-1|>3⇒(x^4+7≤7x^2 )$ or $(2x^3≥8x+5)$ This is what I got for the contrapositive: $∀x∈R,(x^4+7>7x^2 )$ and $(2x^3<8x+5)⇒|2x-1|>5$ or $|2x-1|≤3$ Where would I ...
4
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114 views

How to make a proof your own?

Often, in the interest of time, We look at proofs done by others. It could be on stack, in a textbook, or shown by a friend. In all cases, you don't get the same amount of mental compression of the ...
4
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0answers
94 views

How should you refer to other people's result when writing in math?

Here is what I was taught about coming up with my own results in math school: A lemma builds up to a theorem, which implies corollaries. A theorem you are not proud of is a proposition. However,...
4
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0answers
155 views

Proving that $\mathbb{K}[[x_1,…,x_n]]$ is a UFD

I'm following Mumford's proof in "The Red Book" for proving that a regular local ring is a UFD. He assumes that any $\mathbb{K}[[x_1,...,x_n]]$ is a UFD. I'm trying to find an intuitive, low tech ...
4
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0answers
155 views

Prove that $f_n(x):=\sum_{k=0}^n x^k/k!$ have no real zero for even $n$, and one unique real zero for $n$ odd

I re-write entirely the previous posted (and wrong) proof. I think that this new proof is correct but I need some confirmation. I would like to see too some alternative proof easier than what I did ...
4
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0answers
163 views

Show that $(x_1,x_2)=(0,0)$ is an unstable fixed point for $\dot{x_1} = -x_1 + x_2^6$, $\dot{x_2} =x_2^3 + x_1^6$

Show that $(x_1,x_2)=(0,0)$ is an unstable fixed point for the system $$\dot{x_1} = -x_1 + x_2^6\qquad\dot{x_2} =x_2^3 + x_1^6$$ Hint: Consider the Lyapunov function $V(x_1,x_2) = ax_1^i + bx_2^j$. ...
4
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0answers
89 views

Proving that, for every $\theta \in \mathbb Q$, $\sin(n!\theta\pi)$ has a limit, using $\epsilon$-$N$ definition of limit

If $\theta \in \mathbb Q$, show that $\{\sin(n!\theta\pi)\}$ has a limit How can I rigorously prove that this sequence has a limit using $\epsilon-N$? Is it even possible to do so? I know that we ...
4
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0answers
58 views

Prove that $\mathrm S_m\cdot p=\{p\}\iff p=\sum_{[\alpha]\in \Bbb N^m/\mathrm S_m}p_{[\alpha]}\left(\sum_{\beta\in [\alpha]}X^\beta\right)$

Let $R$ a commutative ring with unity and $\mathrm S_m$ is the set of permutations over $m$ positions, with $m\in\Bbb N$. A polynomial $p\in R[X_1,...,X_m]$ is called symmetric if $\sigma(p)=p$ for ...
4
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0answers
554 views

Derivative of Exponential map on manifolds

I'm trying to compute the derivative of the map $f:\Sigma\times [0,\delta)\to M$ given by $$f(p,t)=\exp_p tN(p),$$ in $X\in T_p\Sigma$, where $(M^n,g)$ is a Riemannian manifold, $\Sigma\subset M$ a ...
4
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0answers
92 views

Recreational math dealing with twin primes

This is kinda recreational math with a goal in mind of progressing further toward a proof of the twin prime conjecture. Consider this: We start with a random prime: $109$ $3*109=327$ $327 \equiv$ ...
4
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99 views

How to prove $\forall x,y\in\mathbb{R}: x^2+y^2 = (x+y)^2 \Leftrightarrow x=0\lor y=0?$

The question I really have is the structure and I am not sure to use pack-unpack or not. Here is my try: Let $x,y\in\mathbb{R}$ Assume $x^2+y^2 = (x+y)^2$ Then $x^2+y^2 = x^2+2xy+y^2$ #by ...
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67 views

Potential Metric on $\Bbb R^n$

As part of a problem set for a course in Metric Spaces. I have to prove that the following is a metric for $\mathbb R ^n $: $$ d_m (\mathbf x , \mathbf y) = \max_{i=1} ^ n | x_i -y_i| $$ Now I am ...
4
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0answers
148 views

How to Prove It, Exercise 6.5.9

Suppose $R$ is a relation on $A$ and $S$ is the transitive closure of $R$. If $(a, b) \in S$, then there is some positive integer $n$ such that $(a, b) \in R^n$, and therefore by the well-ordering ...
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72 views

Much more topology…

These are very similar to my last two questions. I provide them with my thoughts so far: $(1)$ Let $S$ be the collection of all straight lines in the plane which are parallel to the x-axis. If $S$ is ...
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0answers
63 views

Limit of continuous function

Prove or provide a counterexample: 1) $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous. If $(a_{n}) = f(n)$ converges to $L$, then $\lim_{x \rightarrow \infty} f(x) = L$. Counterexample: I ...
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0answers
61 views

Prove this congruence

Let $p$ be a prime of the form $4k+3$ and $m$ an even positive integer less than $p-1$. Prove that $$1^m+2^m+\cdots+\left(\frac{p-1}{2}\right)^m \equiv \left(\frac{p+1}{4}\right)+\left(\frac{p+1}{4}\...
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50 views

Another proof question for real analysis

Let $a, b, c \in \mathbb{R}$. Prove if $a + b = a$ then $b = 0$. Suppose that $a + b = a$. Then $a + b - a = a - a = 0 = b$ by the inverses law for addition. By the Identity law for addition it ...
4
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0answers
679 views

Proving u-substitution the hard way — use only definition of integration with partitions

I am trying to prove integration by substitution, i.e. for $f:[c,d] \rightarrow \mathbb{R}$ continuous and $\phi: [c,d] \rightarrow [a,b]$ continuous on $[c,d]$ and $\mathscr{C}^1$ on $(c,d)$. Then ...
4
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0answers
933 views

Proof for the form of characteristic polynomial

I'd like to proof: The caracteristic polynomial of $A \in M(n\times n, K)$ has the form: $P_A(\lambda) = (-1)^n \lambda^n + (-1)^{n-1} \operatorname{tr}(A)\lambda^{n-1} +\dots +\det(A)$ My proof ...
4
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0answers
112 views

Suppose $G=\{a_1,\dots a_n\}$ is a finite abelian group and $a^2\neq e$ for all nonidentity elements. What is the product $a_1a_2\dots a_n$?

Let $G$ be a finite abelian group such that for all $a \in G$, $a\ne e$, we have $a^2\ne e$. If $a_1, a_2, \ldots, a_n$ are all the elements of $G$ with no repetitions, what is the product $a_1 a_2 \...
4
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0answers
382 views

Transcendence of $e$ (proof)

I'm trying to get through the proof of transcendence of $e$ (the base of the natural logarithm) already for a couple of days, but now I got seriously stuck. Proof is in most sources roughly the same. ...
4
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0answers
453 views

Easy proof of trivial fusion implies normal p-complement

Theorem: Suppose G is a finite group with Sylow p-subgroup P. Then the following are equivalent: The set K of elements of G of order relatively prime to p (the p′-elements) form a subgroup If A and ...
4
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491 views

Theoretical proof of convergence of sequential weight update procedure (Neural Networks and Machine Learning)

My question is at the bottom. (Most of the descriptive words come from Chris. Bishop's Neural Networks for Pattern Recognition) Let $w$ be the weight vector of the neural network and $E$ the error ...
3
votes
0answers
33 views

Prove that matrix $A = \sum_k^{n}x_k\lambda_ky_k^T$

Prove that matrix $\ A = \sum_{k=1}^n {\bf x}_k \lambda_k {\bf y}_k^T.\ $ If $\ {\bf x}_k\ $ and $\ {\bf y}_k\ $ are the corresponding eigenvector from the left and from the right of $\lambda_k$. Any ...
3
votes
0answers
64 views

Prove that the optimal solution of a fitting term does not effect by the outlier

I'm having difficulty in proving the solution of this problem: The given vectors $\mathbf{a}_1$, $\mathbf{a}_2 \in \mathbb{R}^M$ and the variables $\mathbf{b}_{1},\mathbf{b}_2\in\mathbb{R}^M$. ...
3
votes
0answers
101 views

Proof: A tangent space of the manifold of SPD matrices is the set of symmetric matrices

The set of SPD matrices, $\mathbb{P}_n := \{X \in \mathbb{R}^{n \times n} | X=X^T, X \succ 0 \} $, forms a differentiable manifold. Claim: The tangent space at a point, $A, T_A\mathcal{P}_n$ is the ...
3
votes
0answers
37 views

I am writing a formal proof for the first time

I have written a formal proof for the first time and am looking for input on any errors I may have made or general tips thanks. Theorem: For all real numbers x and y, $$x^2 + y^2 + 1 \...
3
votes
0answers
101 views

Proof by induction on an uncountable interval

I'm a little bit confused as to whether or not I can use proof by induction on an uncountable interval. For example, I'm trying to prove that $$(x+y)^n \leq x^n + y^n$$ on the interval $0 \leq n \leq ...
3
votes
0answers
64 views

Is $\sum_{i=1}^{n-1}i^{n-1}\equiv -1\pmod {2n}\Leftrightarrow \text{$n$ is prime }\equiv 3\pmod 4$?

I was looking at the Agoh-Giuga Conjecture, namely, $$\sum_{i=1}^{n-1}i^{n-1}\equiv-1\pmod n\Leftrightarrow n\text{ is prime.}\tag1$$ I decided to see if I could prove it, expecting lots of ...
3
votes
0answers
54 views

In proving that $\sqrt{a}$ is always irrational, $\forall a\in\left\{\Bbb R^+ : 1< a\neq b^2\right\}$… a different way.

I was trying to prove the following statement: $$\sqrt{a}\text{ is always irrational, }\forall a\in\left\{\mathbb{R}^+ : 1<a\neq b^2\right\}.\tag{$b\in\mathbb{Z}$}$$ I know there is at least one ...
3
votes
0answers
32 views

Relations Proof: Intersection of Two Sets of Path Length n

I am trying to answer the following question: Prove that $(R \cap S)^{n} \subseteq (R^{n} \cap S^{n})$ for all $n\geq1$ Attempt using Induction Base Step $(R \cap S)^{1} \subseteq (R^{1} \cap S^{1}...