For questions about the writing of proofs. But instead of focusing the mathematics behind the proof, these questions ask about the details and implementation of the proof.

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1
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1answer
69 views

$\mathbb{A}^2\setminus (0,0)$ is not affine

I want to prove that $X = \mathbb{A}^2\setminus (0,0)$ is not affine. My attempt: If $\Bbbk[X] = \Bbbk[x,y]$ then $X$ is not affine since $(x,y) \subset \Bbbk[x,y]$ is a proper ideal, but $V(x,y) \cap ...
1
vote
1answer
27 views

On existence of square root of positive elements of a unital $C^*$-algebra

Given a unital $\mathcal{C}^*$-algebra $A$ and a positive element $a \in A$, I am trying to prove the existence of a square root $a^{\frac{1}{2}}$ i.e. a positive element $b \in A$ such that $b^2 = ...
0
votes
1answer
49 views

Prove that length of a curve is finite / infinite

I have problems proving the following: Let $\alpha > 0 $. Consider the curve $\gamma : [0,1] \to \mathbb{R}^2$ given by $\gamma (0)=(0,0) , \gamma(t) = (t^\alpha \cos ( \frac {1}{t}), t^\alpha ...
1
vote
1answer
28 views

Converging subsequences and subsets having infinite elements

We have a metric space (V,d). Proof that the following two properties are equivalent. a) Every sequence $a_n \in V$ has a subsequence which converges to a element $x \in V$ b) For every ...
0
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1answer
36 views

Proving that a solution exists

Proof that there exists a $x>0$ with $x \in \mathbb{R}$ s.t. $\sin(x) = \frac{x}{2}$ I tried to use the intermediate value theorem, but I don't know how to apply it correctly. Obviously ...
0
votes
4answers
47 views

How can I prove that this matrices statement is false?

How can I prove that this is not true: If for matrices A, B and C, AB=AC and A is not the zeroth matrix, then B=C.
3
votes
1answer
27 views

Characterization of subsets of $\mathbb{R}^n$ of the form $X+Y$

The following comes from the mathematical tripos exam at Cambridge: Let $X,Y \subset \mathbb{R}^n$, and define $X+Y = \{x+y : x \in X, y \in Y\}$ Prove or disprove each of the following: (i) If ...
2
votes
2answers
40 views

Prove using Hilbert calculus $\forall x(Px\rightarrow x\equiv a)\vdash Pb\rightarrow Pa$, formal proof.

Prove: $\forall x(Px\rightarrow x\equiv a)\vdash Pb\rightarrow Pa$ Using Hilbert Calculus Format of solution: Step (my understanding) Solution: (1) $\forall x(Px\rightarrow x\equiv a)\vdash ...
3
votes
2answers
71 views

Proving that $\sum_{i=2}^n(5i-4)=\frac{n(5n-3)-2}{2}$ for all $n\geq 1$ by mathematical induction

I have this question: Show, using mathematical induction, that for all natural numbers $n$, $$6 + 11 + 16 + 21 + \cdots + (5n-4) = \frac{n(5n-3)-2}{2}$$ I am confused in that that question states ...
2
votes
4answers
38 views

Does an arbitrary matrix $X \in M_{n \times p}$ have a SVD?

I have proven, as below, that if $X \in M_{n \times n}$ is symmetric, then it has a SVD. $D(\lambda_i) = \text{Diag}(\lambda_i)$ is a diagonal matrix with entries $\lambda_1, \lambda_2, \dots$. ...
2
votes
1answer
39 views

Using lipschitz estimate to show $|f_n(x) - f_p(x) - (f_n(c)-f_p(c))| \leq |b-a|\sup_{y \in (a,b)}|f'_n(y)-f_p'(y)|$

Assume $(f_n)$ is a sequence of functions that are continuous on $[a,b]$ and differentiable on $(a,b)$. Then using Lipschitz estimate to prove that $$|f_n(x) - f_p(x) - (f_n(c)-f_p(c))| \leq ...
5
votes
1answer
33 views

Proof check, showing pointwise convergence

My problem is this: For $x \in [0,\frac{\pi}{2}]$, $f_n(x) = \frac{nx}{1 + n\sin(x)}$ Find the pointwise limit of $(f_n)$ for all $x \in [0, \frac{\pi}{2}]$ I am not sure if the way I constructed ...
0
votes
0answers
14 views

Prove that unitary normal vector to a manifold is $\nabla g / |\nabla g|$ [duplicate]

I want to solve the following exercise: Let $A\subset\mathbb{R}^n$ open $g:A\to\mathbb{R}$ of class $C^{1}$ and $g'(x)\not=0$ in each $x\in A$ then I want to compute $dV$ in the differentiable ...
1
vote
2answers
39 views

Let $V$ be a $K$ - Vector Space, $f: V \rightarrow V$ a linear map. Under some condition, show that $v, f(v),…,f^n(v)$ is linear independent.

Let $V$ be a $K$ - Vector Space, $f: V \rightarrow V$ a linear map. Let $v \in V$. May a number $n ≥ 0$ exist, so that: $f^n(v) \not= 0$ and $ f^{n+1}(v) = 0$. Show that $v, f(v),...,f^n(v)$ is ...
2
votes
1answer
24 views

Prove that if $f$ is an invertible function and $g$ is an inverse, then the codomain of $g$ is equal to the domain of $f$ and vice versa

I am trying to show, without using the bijection properties, what is above. Assume $f$ is an invertible function and $g$ is an inverse of $f$. For $f \circ g $ to be well defined then the image of ...
0
votes
1answer
19 views

Placement of quantifiers in a symbolic statement

I have the statement: Let $A_n$ be an indexed set of numbers defined by $A_n = n\mathbb{Z}_{\geq m}$ for $n,m \in \mathbb{Z}$. Consider the claim C: For all $n$, if $x,y$ is in $A_n$, then ...
-2
votes
2answers
36 views

Help with Discrete Structures proof [closed]

I don't have any clue how to do this $\sum\limits_{i=1}^n(-1)^{i-1}i^2 = (-1)^{n-1}n(n+1)/2$ whenever n is a positive integer. Please explain each step.
1
vote
2answers
42 views

Proving the well ordering principle

THe well ordering principle has that every subset of $\mathbb{Z}^+_0$ has a least element. or if $S$ is a non-empty subset of $\mathbb{Z}^+_0$ and $S = \{a_1, a_2, a_3 ... a_n\}$, then there is a ...
1
vote
2answers
44 views

Proof variance of Geometric Distribution

I have a Geometric Distribution, where the stochastic variable $X$ represents the number of failures before the first success. The distribution function is $P(X=x) = q^x p$ for $x=0,1,2,\ldots$ and ...
0
votes
1answer
52 views

Contradiction proofs

I'm a first year Physics student and I have some trouble approaching Proofs by Contradiction in some of my Math classes. Once I get the first 2 or 3 statements I can finish the proof but a lot of the ...
2
votes
0answers
92 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 ...
2
votes
1answer
40 views

Why this set is the pole set of $z$?

Suppose $V\subset \mathbb P$ a projective variety, $z$ a rational function on $V$ and let's define $J_z=\{F\in k[X_1,\ldots, K_{n+1}\mid \overline Fz\in \Gamma_h(V)\}$ and $S_z$ the polo set of $z$, ...
2
votes
5answers
48 views

Prove that $X \subset Y \implies f^{-1}(X) \subset f^{-1}(Y)$

Let E and F be two sets and $f: E \to F $ be a function, and $X, Y \subset F$. Prove that $X \subset Y \implies f^{-1}(X) \subset f^{-1}(Y)$ My answer: Let $y \in X$, then $f^{-1}(y) \in ...
2
votes
1answer
32 views

Injection proof

Prove that if f is injective, then $f(A \cap B) = f(A)\cap f(B)$ My answer: i) $f(A \cap B) \subset f(A) \cap f( B )$ Take an $x \in A \cap B$. $x \in A \cap B \implies x \in A \land x \in B$ $x ...
1
vote
1answer
25 views

Prove $\vdash\forall x(\alpha\to\beta)\to(\exists x\alpha\to\exists x\beta)$

I want to show that: $\vdash\forall x(\alpha\to\beta)\to(\exists x\alpha\to\exists x\beta)$ I started my deduction as follows: $\vdash\forall x(\alpha\to\beta)\to(\forall ...
9
votes
2answers
156 views

Proving $\sqrt{2}(a+b+c) \geq \sqrt{1+a^2} + \sqrt{1+b^2} + \sqrt{1+c^2}$

I've been going through some of my notes when I found the following inequality for $a,b,c>0$ and $abc=1$: $$ \begin{equation*} \sqrt{2}(a+b+c) \geq \sqrt{1+a^2} + \sqrt{1+b^2} + \sqrt{1+c^2} ...
-3
votes
3answers
51 views

Prove that for any sets $A$ and $B$ there is a unique set $C$ such that $A ∆ C=B$

Using Venn diagram, I see that letting $C = A ∆ B$ works. But I have trouble proving this using notations. Show me how to do the existence and uniqueness part of this proof.
13
votes
7answers
199 views

What are statements about the natural numbers where induction is impossible or unnecessary to prove?

I'm looking for statements like "for all natural numbers, ____" where induction would be impossible or unnecessarily complicated. This is for pedagogical reasons. When students first learn induction, ...
-2
votes
2answers
65 views

Proving that $A\cup\emptyset=A$ and $A\cap\emptyset=\emptyset$ [closed]

I need to prove that $A\cup\emptyset=A$, and $A\cap\emptyset=\emptyset$. It's seem like it's obvious, yet how can I prove it mathematically?
0
votes
1answer
52 views

The Change-making problem algorithm proof (at the dynamic programming method)

I saw here the algorithm for the "Change-making problem" (at the dynamic programming method). I saw it here: http://www.columbia.edu/~cs2035/courses/csor4231.F07/dynamic.pdf I'm trying to find a ...
1
vote
3answers
102 views

Prove $\frac{1}{n} =\frac{1}{n+1}+\frac{1}{n(n+1)}$ for all integers $n\in\Bbb Z$

I'm pretty sure that we need induction, since it's the format I had to use for previous problems similar to this (it isn't specified that it HAS to be an inductive proof, either, if there is another ...
0
votes
1answer
19 views

If $G$ is a connected graph with $n$ vertices and$ n - 1$ edges then $G$ is a tree, using Induction.

I am still new to proof methods and not sure if this is the correct use of induction. Base case: $n = 1$ has $0$ edges and is a tree. Assume every connected graph with $k$ vertices and $k-1$ edges ...
0
votes
1answer
58 views

Some proofs regarding Stirling numbers

I would like you to help me to prove two proofs correlated with Stirling numbers (the first one includes Stirling numbers of the second kind and the second one I guess Stirling numbers of the second ...
2
votes
3answers
30 views

Base of the $\mathbb{R}$ vector space that contains all real functions: $f(x) \not= 0$ for finitely many x $\in\mathbb{R}$

I did already prove that this is a vector space. It is easily shown that addition and scalar multiplication with functions that hold the above property again yields a function with $f(x) \not= 0$ for ...
1
vote
1answer
33 views

An equation to prove with terms of Fibonacci sequence

I would like to prove an equation but I have stuck. The equation that is to prove is the below: $f(n)^2 + (-1)^{n+1} = f(n+1)f(n-1) , n \ge 2$. I'm trying to do an inductive proof of this equation. ...
3
votes
2answers
81 views

Theorem 7.2 in General Topology by S. Willard

Theorem 7.2 If $X$ and $Y$ are topological spaces and $f:X \to Y$ , then the following are all equivalent :- I) $f$ is continuous. II) for each E $\subset X$ , $f(\bar E) \subset ...
1
vote
1answer
16 views

Proving that in a complete graph $\lambda(K_{n}) = \delta(K_{n})$

Since $K_{n}$ is n-1 regular. Then $\lambda(G)$ must be n-1. Since $\lambda(K_{n}) \leq \delta(K_{n})$ then by definition they must be equivalent. Can I use the definition or should I say since ...
1
vote
1answer
20 views

Proving that in a complete graph $\kappa(K_{n}) = \delta(K_{n})$

Since $K_{n}$ is n-1 regular. Then $\delta(G)$ must be n-1. Since $\kappa(K_{n}) \leq \delta(K_{n})$ then by definition they must be equivalent. Am I approaching this proof the right way?
3
votes
3answers
100 views

Prove that if $B$ is similar to $A$, then $B^T$ is similar to $A^T$ .

If two matrix ($A$ and $B$) are similar if there exists an invertible matrix $P$, such that: $$ B=P^{-1} A P $$ I'm thinking if I can prove that $A$, $B$ , $A^T$ and $B^T$ have the same characteristic ...
0
votes
2answers
30 views

Understanding Proof that $\mathbb{R} \setminus A$ is dense. Verify proof.

Here's the proof I was given but with two minor? differences Proposition.- If $A$ is countable then $\mathbb{R} \setminus A $ is dense. Proof: Suppose otherwise, then there exists real numbers $a$ ...
1
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0answers
27 views

Developing proof writing and logical skills

What resources can a person turn to in order to develop their proof writing and logical skills? The advanced calculus course I'm taking has made me realize how weak my logic and proof writing skills ...
5
votes
3answers
53 views

If $a\in\mathbb{Q}$, prove that the sequence $\{\sin(n!a\pi)\}_{n=1}^\infty $ has a limit.

This exercise is from Methods of Real Analysis by Richard Goldberg. If $a\in\mathbb{Q}$, prove that the sequence $\{\sin(n!a\pi)\}_{n=1}^\infty$ has a limit. I think this proof relies on the ...
1
vote
2answers
37 views

Understanding line of given proof

I have to understand a set of proofs and I don't understand the reasoning behind this line "This is an injection, if $g(b_1) = g(b_2)$ then $F_{b_1}$ And $F_{b_2}$ intersect, which we shown never ...
0
votes
1answer
27 views

Proof that the sup of a set is the least upper bound.

I have to prove: Assume $s \in \mathbb{R}$ is an upper bound for a set $A \subseteq \mathbb{R}$. Then, $s = \text{sup}A$ if and only if, for every choice of $\varepsilon > 0$, there exists an ...
2
votes
1answer
50 views

Proving $R(3,4)\le 9$

I am trying to prove $R(3,4)\le 9$. This is my approach: For any $K_9$ we have (WLOG) at least 4 red edges by the pigeonhole principle. Consider all of the edges between these 4 red edges, if ...
-1
votes
2answers
18 views

On Closure of Product subset of $\Bbb R×\Bbb R$

Suppose that $\Bbb R×\Bbb R$ has the standard topology. If $S=\left\{(t,\sin{\frac{1}{t}})\mid t\in R\text{ and }t\gt 0\right\}$. Show that $(0,0)$ $\in \overline{S} $
0
votes
2answers
60 views

$\mathbb{R}$ is uncountable with Cantors Diagonal argument (how to improve binary expansion specificity?)

I know it's spelled out more than usual, but this is an introduction to higher math class. If there's any way I can improve this, please let me know. Thank you in advance. Let ...
0
votes
1answer
34 views

Prove that $f^{-1}\left(U_1\times\cdots\times U_\kappa\right)=\bigcap_{i=1}^\kappa \left(f_i\right)^{-1}\left(U_i\right)$

Im working through Bloch's Proofs and Fundamentals and exercise 4.3.11 is Let $B$ be a set, let $A_i,\cdots,A_\kappa$ be sets for some $k\in\mathbb{N}$ be a subset for all ...
0
votes
0answers
73 views

Herbrands Algorithms and greek philospher

So the problem states "outline the steps in Herbrands algorithm leading to the proof that the following statement is right. ...
1
vote
3answers
18 views

Proving continuity with two different metrics

Problem statement: Let $X$ be the set of all continuous functions $f:[a,b]\rightarrow \Bbb R$, and define the metric $d^*(f,g)$ on $X$ by $$d^*(f,g) = \int_{a}^{b} |f(t) - g(t)|dt$$ Now, for each ...