For questions concerning a specific proof, asking for verification, identifying errors, suggestions for improvement, etc.

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0
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1answer
35 views

Ordered Field $\mathbb{F}$ Corollary Proof

I wanted to check my proof for a corollary on ordered fields $\mathbb{F}$. Here is the corollary: Corollary: Let $\mathbb{F}$ be an ordered field and $a\in\mathbb{F}.$ If $a>0$, then ...
2
votes
1answer
44 views

Cantor set's endpoints.

Prove that: If $[a,b]$ is one of the closed intervals that makes up one the approximation $C_k$ of the Cantor set then the endpoints $\{a,b\}\subset C$ where $C$ is the cantor set. I should prove ...
0
votes
4answers
50 views

Expectation of non-negative random variable

Let $X$ be a non-negative random variable. In a proof for $E[X]=\int_0^\infty P(X>t)dt$ from the answer of this question, we use Fubini for the middle quality. Why do we need $X$ to be ...
3
votes
1answer
40 views

Radius of convergence, prove that $\sum\limits_{n=0}^{\infty} a_n z^n$ converges absolutely for every $z \in \mathbb{C}$ with $|z| < R$

A proof of this is given in my lecture notes as follows: We define $R$ to be $\sup \{|z| \in \mathbb{R} : \sum |c_k z^k|$ converges $\}$ when the supremum exists. Prove that $\sum |c_k ...
0
votes
1answer
22 views

Steps involved when showing this induced map on homology is welldefined

I am showing that $H_0(X,R)=R$ when $X$ is a path-connected topological space. Let the zero boundary map be $\partial_0 : C_0(X) \to R$, $c \mapsto 0$. Define a map $\varphi : C_0(X) \to R$ by ...
5
votes
1answer
65 views

Is this an accurate proof that no perfect square is of the form $4k+3$? ($k$ an integer)

A positive integer $n$ is a perfect square. Prove that it cannot be of the form $4k+3$, where $k$ is an integer. I tried to prove this by proof by contradiction: if $n$ is a perfect square, then ...
0
votes
1answer
37 views

How can I prove that the zeroes of $f(z)=1+1/2^z$ have no real part?

I want to prove that the zeroes of the function $f(z)=1+1/2^{z}$ have no real part. Is the following correct? $f(z)=0$ so $2^{z} = -1$ and $-1=e^{i\pi}$ so $e^{i\pi} = e^{z\ln2}$ therefore $z= ...
0
votes
0answers
38 views

How do I show $G_0$ and $G_1$ are conjugate subgroups? Please improve my answer.

Is my solution below correct? Please read through it and tell me if it seems complete or to make sense. Question: Let $E$ be path-connected. Let $p : E → B$ be a covering map and $p_∗$ be the ...
1
vote
1answer
17 views

Prove center of a group is a subgroup using one-step subgroup test

I'm not sure if this is correct. It doesn't seem so. If $a,b \in C$, then we must show $ab^{-1} \in C$. $$ab^{-1}x=axb^{-1}=xab^{-1}$$ This doesn't seem correct. I've seen two-step subgroup tests ...
2
votes
0answers
62 views

A more detailed, rigorous proof that a suspension space is not necessarily contractible

Is my answer/proof correct? Please help me make my proof more rigorous and detailed. I need everything to be absolutely clear. Question: Let $X$ be a topological space. The suspension of $X$, ...
6
votes
1answer
67 views

Proving the Cone is Contractible: Is my Proof correct?

Is my answer/proof correct? Please help me make my proof more rigorous and accurate. I need everything to be absolutely clear and rigorous. Thank you. Question: Let $X$ be a topological space. The ...
2
votes
2answers
53 views

Proof : If $f$ continuous in $[a,b]$ and differentiable in $(a,b)$ and there is $c \in (a,b)$ so $(f(c)-f(a))(f(b)-f(c))<0$

I need to proof this : If $f$ continuous in $[a,b]$ and differentiable in $(a,b)$ and there is $c \in (a,b)$ so $(f(c)-f(a))(f(b)-f(c))<0$ then there is $d \in (a,b)$ so $f'(d)=0$. I'm not sure ...
3
votes
1answer
77 views

Is my Proof Correct and Rigorous: Proving that Quotient Space is Hausdorff

Question: Let $X$ be a topological space and let $A ⊂ X$. Define an equivalence relation $∼$ on $X$ such that the equivalence classes are: • $A$ itself, and, • Singletons {$x$} such that $x /∈ A$. ...
2
votes
3answers
41 views

Why can't a direct proof be made backwards?

Say we have the following implication: $$\textit{Let $x\in \mathbb{Z}$. If $5x-7$ is even, then x is odd. }$$ The method used by my book to prove this implication is by means of a proof by ...
2
votes
1answer
39 views

Proposed proof for convergence in Sobolev space

Consider the Anisotropic Sobolev Space defined by: $$W^{1,\overrightarrow{p},\epsilon}(\Omega) := \{ u \in L^{1+\frac{1}{\epsilon}}(\Omega), \frac{\partial u}{\partial x_{i}} \in L^{p_{i}}(\Omega), ...
1
vote
2answers
57 views

About the infinitude of some kind of primes? [on hold]

I will propose this proof: A Mersenne number always has the form $$2^{p}-1=4n+3$$ since for all $p≥2$ we have $$2^{p}-1≡-1(mod4)≡3(mod4)$$ The Dirichlet prime number theorem ...
1
vote
1answer
30 views

Proof on $\bigcup A=\varnothing\Rightarrow\forall a\in A, a=\varnothing$

Is this proof valid? $\textbf{Claim: }\bigcup A=\varnothing\Rightarrow\forall a\in A, a=\varnothing$ Proof. Let us suppose that there was an $x\in A$ where $x\neq\varnothing$. Since $x\in \bigcup A ...
1
vote
2answers
75 views

Don't understand proof that $x_n \rightarrow A$ $\iff$ every subsequence of $\{x_n\}$ converges to $A$

So we are given that for all $\epsilon > 0$, there exists an $N$ such that for all $n \geq N$, we have that $|x_n - A| < \epsilon$ and we want to show that for all $\epsilon' > 0$, there ...
1
vote
1answer
49 views

A question on Artinian and Noetherian rings.

All rings are commutative and unital. Suppose that $A$ is a ring in which the zero ideal can be written as a product of maximal ideals of $A$. I try to prove that $A$ is Noetherian if and only if ...
2
votes
1answer
27 views

Show this function is onto

I gave a mapping A to C such that A is the set of left cosets in G described as $A$={$N(H), gN(H),...g_nN(H)$} for N(H) is the normalizer of H in G and C is the set of conjugates of H, ...
-3
votes
1answer
45 views

fake proof of $\forall a. \forall b. a = b \to 1 = 0$

I saw a less formal version of this fake proof that claimed to prove $2=1$ but because it assumed $a=b$ from the start I knew why it was wrong. It does seem however that the proof can be used to prove ...
3
votes
2answers
43 views

Differential Equation: $\frac{\mathrm{d} y}{\mathrm{d} x} = xy + y\sin x$

I'm trying to solve this differential equation and believe I may have solved it using the "separable equations" method. Here's my work: $$\frac{\mathrm{d} y}{\mathrm{d} x} = xy + y\sin x = y(x + ...
0
votes
1answer
32 views

Prove that if $\: \forall_{t \in \mathbb{R}} \: \overline{\overline{A}}_t = c$ then $\overline{\overline{\bigcup_{t \in \mathbb{R}} \: A_t}} = c$

Prove that if $\: \forall_{t \in \mathbb{R}} \: \overline{\overline{A}}_t$ is equal to cardinality $c$, then $\:\overline{\overline{\bigcup_{t \in \mathbb{R}} \: A_t}}$ is also equal to cardinality ...
1
vote
1answer
26 views

Am I doing this approximation correctly? (least squares method)

Here is the problem. Find the function $f$ of the type $f(x) = a\cos x + b\sin x$ which best approximates the function $g$ in the points : $$ \begin{array}{ c | c | c | c | c | c | c } x & ...
2
votes
0answers
41 views

Is this article about exponential wrong?

Wikipedia : http://en.wikipedia.org/wiki/Formal_power_series Assume that the ring R has characteristic 0. If we denote by exp(X) the formal power series $exp(X)=1+X+\frac{X^2}{2} + ...
2
votes
0answers
27 views

Intesection point of feet of altitudes

If triangle has vertexes at $(x_1,y_1),(x_2,y_2),(x_3,y_3)$, is the intersection points of feet of altitudes $$x_h = \frac{x_1x_2(y_2-y_1) + x_2x_3(y_3-y_2) + x_3x_1(y_1-y_3) + y_1^2(y_3-y_2) + ...
3
votes
0answers
27 views

Showing points of continuity of a function f(x) that takes the value 1/n whenever x belong to a sequence {An} and is zero elsewhere.

I am given a sequence $(An), n=1,2,3,...$ which consists of distinct numbers, which converges to $3$ as $n$ tends to infinity, but none of its terms are equal to $3$. Then I am given a function $f(x) ...
1
vote
1answer
33 views

bounded intervals and partitions

Can you please check my proof? Question Let $I$ be a bounded interval of the form $I = (a, b)$ or $I= [a, b)$ for some real numbers $a< b$. Let $I_1, I_2, ..., I_n$ be a partition of $I$. ...
0
votes
0answers
14 views

Every Solution set of Homogeneous system is a linear combination of fundamental solutions

Prove: Every Solution set of Homogeneous system is a linear combination of fundamental solution. can I say that the fundamental solution is a trivial basis therefore is spans the Null space? can I ...
1
vote
0answers
36 views

Proof of complex Bolzano–Weierstrass Theorem

Here is the proof from my lecture notes: Write $z_n = x_n + i y_n$. Let $M$ be such that $|z_n| \leq M$ for all $n$. Then by the definition of modulus in $\mathbb{C}$, we have $|x_n| \leq M$ and ...
1
vote
0answers
14 views

Showing if $\lim_{n\to\infty} a_n=L$ then $\lim_{n\to\infty} -2a_n=-2L$ using defintion

If $\displaystyle \lim_{n\to\infty} a_n=L$ then prove using the limit definition that: $\displaystyle \lim_{n\to\infty} -2a_n=-2L$. From the given and the definition we know that: ...
2
votes
3answers
37 views

Proving if $A \subseteq B$ and $A \nsubseteq C$ then $B \nsubseteq C$

This is one of the problem I have been solving in Velleman's How to prove book: Prove that if $A \subseteq B$ and $A \nsubseteq C$ then $B \nsubseteq C$ This is my solution: Suppose $A ...
4
votes
1answer
95 views

Cutting a pie with a fork

You baked a pie in the shape of a disc, with some cherries spread unevenly on its top. You want to give each of your two children a piece of cake such that: The pieces are congruent - have the same ...
4
votes
1answer
31 views

Partial Converse of Holder's Theorem

Holder's Theorem is the following: Let $E\subset \mathbb{R}$ be a measurable set. Suppose $p\ge 1$ and let $q$ be the Holder conjugate of $p$ - that is, $q=\frac{p}{p-1}.$ If $f\in L^p(E)$ and $g\in ...
1
vote
2answers
126 views

$\displaystyle\lim_{n\to\infty}a^{n}\left(\frac{n+1}{n}\right)^{n^{2}}$ converges for $a$ in what range?

$\displaystyle\lim_{n\to\infty}a^{n}\left(\frac{n+1}{n}\right)^{n^{2}}$ converges for $a$ in what range? I tried $\displaystyle\lim_{n\to\infty}\ln ...
0
votes
2answers
39 views

Prove that $\frac{1}{2}ab \equiv \int_0^b \! f(x) \, \mathrm{d}x$ when calculating the area of a right triangle.

Triangle $ABC$ is a right triangle with sides $AB$, $BC$ and $AC$. $a$ is the length of $AB$. $b$ is the length of $BC$. $c$ is the length of $AC$. If $a = 3$, and $b = 4$, we can use ...
1
vote
0answers
32 views

Proving that a certain local ring is regular

I understand that this is a special case of the Jacobian criterion, but I was hoping that there was a simpler argument to prove it than for the criterion itself (I don't fully understand the proof of ...
2
votes
1answer
35 views

What is the dimension of $A-B$, where $B$ is a subspace of $A$?

My question is really simple, what is the dimension of $A-B$, where $B$ is a subspace of $A$? this space is well-defined? I found this space in this paper on page 440: Following my calculations in ...
3
votes
2answers
29 views

How to prove that if R is a partial order then also $R^{-1}$ is also a partial order?

My guess was to divide the problem into 3 cases: prove that $R^{-1}$ is reflexive, antisymmetric and transitive. To prove that $R^{-1}$ is reflexive, I did: If $R$ is reflexive, that follows that ...
1
vote
0answers
22 views

Proving with definition if $\lim_{n\to \infty}a_n=\infty$ then $\lim_{n\to \infty}-a_n=-\infty$

Prove by definition that: Let $a_n$ be a sequence such that, if $\displaystyle\lim_{n\to \infty}a_n=\infty$ then $\displaystyle\lim_{n\to \infty}-a_n=-\infty$. From the defintion, if ...
0
votes
0answers
32 views

Transition matrix of a double induced Markov chain

Here is how we defined induced Markov chains: Suppose that $(X,E,P)$ is an irreducible Markov chain, where $X=(X_i)_{i\in\mathbb{N}_0}$, $E$ is the state space and $P=(p_{i,j})_{i,j\in E}$ is the ...
3
votes
4answers
57 views

A problem with proving using defintion that $\lim_{n\to\infty}\frac {n^2-1}{n^2+1}=1$

Prove using the definition that: $$\displaystyle\lim_{n\to\infty}\frac {n^2-1}{n^2+1}=1 $$ What I did: Let $\epsilon >0$, finding $N$: $\mid\frac {n^2-1}{n^2+1}-1\mid=\mid\frac ...
0
votes
1answer
34 views

Some little proofs (example: if $y(y-6)/3=x$ then $x\ge3$) [closed]

I have here some little proofs, which I made as an preperation for an exam and I would like to ask if they are right. Let $x,y ∈ ℝ $. Prove that if $\frac{y(y-6)} {3}$ $= x$ then $x ≥ -3$. Proof ...
1
vote
2answers
29 views

Proof on elementary set theory

I want to show that $A\times(A\setminus B)=(A\times A)\setminus(A\times B).$ So I started off with $(A\times A)\setminus(A\times B)=\{(a,b):(a,b)\in(A\times A) \wedge (a,b)\notin(A\times ...
4
votes
1answer
76 views

Suppose that $f: A \to B$ and $g: B \to C$ are functions.

Suppose that $f: A \to B$ and $g: B \to C$ are functions. Prove the following: (a) If $g \circ f$ is injective, then $f$ is injective. Proof. Assume that $f$ is not injective. Then ...
2
votes
0answers
49 views
+50

Induced Markov chain - verify Markov property and another property

First, here is how we defined induced Markov chains: Suppose that $(X,E,P)$ is an irreducible Markov chain, where $X=(X_i)_{i\in\mathbb{N}_0}$, $E$ is the state space and $P=(p_{i,j})_{i,j\in E}$ is ...
1
vote
1answer
41 views

Isomorphism - Linear Algebra ( someone check if my work is enough please)

I have a linear transformation $T: P_3\to \Bbb{R}_4$ defined by a matrix $A$ To show that $T$ is an isomorphism, is it enough to show that $T$ is a bijection by using $A$ to show that it is ...
0
votes
0answers
27 views

Noetherian ring and radical

Show that in a Noetherian ring $I$ and $J$ have the same radical if and only if there is a positive integer $N$ such that $I^N \subset J$ and $J^N \subset I$. [Hint: for the ``if'' direction, use a ...
0
votes
3answers
45 views

Help explain the end of this proof for infinitely many primes?

by contradiction, assume finitely many primes $p_1, p_2,\cdots, p_k$. let $N = p_1p_2\cdots p_k + 1$. Note $N > 1$. Now, by the fundamental theorem of arithmetic, there exists a number $p_j$, where ...
0
votes
0answers
31 views

noetherian Ring

Let $I$ be an ideal in a Noetherian ring $R$. Prove that there exists a positive integer $N$ such that $(rad(I))N ⊂ I$. [Hint:Let $rad(I)=⟨g_1,...,g_k⟩$,and suppose $g_i^{n_i} ∈I$.Use ...