Questions tagged [proof-explanation]

This tag is for readers who ask for explanation and clarification of some steps of a particular proof.

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in a commutative Artinian ring, prime ideals are maximal

Question Prove that: in a commutative Artinian ring, prime ideals are maximal. I need some help understanding the proof. The setting is a commutative Artinian ring, so this ring satisfies the D.C.C on ideals. Here's the proof I'm trying to…
Ninosław Brzostowiecki
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If $X$ is a metric, then $X$ is compact if and only if $X$ is sequentially compact - axiom of choice usage

I'm going through a proof for the theorem: If $X$ is a metric, then X is compact if and only if X is sequentially compact. I have already posted this here. However this time I'm looking at the converse. The proof so far is this: We prove the…
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Proving the multiplication formula of Gamma function

Evaluate this integral $$\int_{0}^{\infty} \frac{x^{2m}}{1+x^{2n}}dx$$ then use the result and the relationship between gamma and beta functions to prove that $$\Gamma({x})\Gamma(1-x)= \frac{\pi}{sin(\pi x)}$$ I was able to evaluate the integral…
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How to prove that doesnt exist a natural number such that is equal to it successor from Peano axioms?

Im getting a hard time trying to prove the general for any natural number $n$ such that $$\nexists n\in\Bbb N: S(n)= n$$ From the second Peano axiom we know that $$\nexists n\in\Bbb N: S(1)= n$$ and from the third axiom that $$S(a)=S(b)\to a=b$$ I…
Masacroso
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Space-Hierarchy Theorem in Theoretical CS

Sipser has a proof this theorem that goes like this: $$D = \text{"On input } w$$ $$1. \text{Let } n \text{ be the length of } w$$ $$2. \text{Compute } f(n) \\ \text{using space constructibility and mark off this much tape. If later stages ever…
lars
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How do you find the Inverse Laplace transformation for a product of $2$ functions?

If $$\mathscr{L}(y)=\frac{ne^{-pt_0}}{n^2+\omega^2}\left(\frac{1}{p+n}+\frac{n}{p^2+\omega^2}-\frac{p}{p^2+\omega^2}\right)$$ show that $$\bbox[yellow] {y=n\left(\frac{e^{-n(t-t_0)}}{n^2+\omega^2}+\frac{n\sin\Big(\omega (t -…
BLAZE
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Why a dimension of a vector space $V$ is equal the sum of multiplicity algebraic of eigenvalues?

I read the proof in a book, but I don't understand. Can someone help me to understand, please? Let $T: V \longrightarrow V$ a linear operator, $V$ a vectorial space with finite dimension, $\lambda_i$ different eigenvalues of $T$ with $i \in…
George
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Some details in the proof to show that a reciprocal function is continuous.

Let $(X,d)$ be a metric space and $f:X\to\mathbb{R}$ continuous. Let $Y=\{x\in X:f(x)\neq0\}$. Prove in detail that the function $g:Y\to\mathbb{R}$ defined by $g(x)=\frac{1}{f(x)}$ is continuous. Proof: Fix $x\in Y$. Let $f(x)=r>0$ (the proof is…
user71346
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Clarification on proof of the sum of Euler $\phi$ fcn: $\sum_{d|n}\phi(d)=n$

In the proof, it says "clearly equals $\phi(n_1)$". I don't see how this is clear. I also don't see how this implies $n=\sum_{d|n}\phi(n/d)$. Can someone please clarify this proof? (from A Course in Combinatorics book by van Lint/Wilson, page…
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Setting up a Problem for Induction that Involves Inequality and a Function

The problem states: Use mathematical induction to prove that any stack of ($n\geq 5$) pancakes can be  sorted using at most $2n-5$ flips.  You may use the fact  any stack of $5$ pancakes can be  sorted with at most $5$ flips I am trying to set it up…
StevenOL
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Proving every infinite set $S$ contains a denumerable subset

I have some trouble understanding this proof of every infinite set $S$ contains a denumerable subset contained in Charles Pugh's Real Analysis text. Proof: Since $S$ is infinite, it is nonempty and contains $s_1$. Then $S \backslash \{s_1\}$ is…
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How can I use the principle of least integer to prove that a non-empty finite set of non-negative integers has a maximum element?

Principle of least integer: Every non-empty set of non-negative integers has a minimum element. How can I use this principle to prove that a non-empty finite set of non-negative integers has a maximum element?
vivek kumar
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Identifying the constant of integration

A short extract from a book of mine states that: If $$\color{red}{A(x,y)\frac{\partial p}{\partial x}+B(x,y)\frac{\partial p}{\partial y}=0\tag{A}}$$ where $p=p(x,y)$ and $A$ and $B$ are also functions of $x$ and $y$. Also $$\color{blue}{\rm d…
BLAZE
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The Hadamard determinant problem: understanding a proof for the upper bound on matrix determinants

I'm working through a proof on The Hadamard determinant problem which can be found in Proofs from THE BOOK. I don't understand how the transition from real valued matrices $A$ with entries in $\{-1,1\}$ to matrices $B = A^T A$ is justified. Of…
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Prove that a saddle point for the Lagrangian function - then it's a global min for constraint problem

I have to prove: If $(x,u,v)$ is a saddle point for the Lagrange function then $x$ is a global minimum point for the corresponding constrained problem. I tried to find hints on the web -- without success. I started with the defintion of a…
Kenni
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