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

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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
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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
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4answers
58 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
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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
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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
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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
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0answers
53 views

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
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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 ...
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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
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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
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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 ...
3
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1answer
32 views

Rational Root (algebra)

I am trying to prove the following statement: Let $f(x)=a_nx^n+···+a_0$ be a polynomial of degree $n$ with integer coefficients. Suppose $r/s∈Q$ is a root of $f$ where $\gcd(r,s) = 1$. Prove that ...
12
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3answers
155 views

Prove that $\frac{(p^{n}-1)(p^{n}-p)…(p^{n}-p^{n-1})}{n!} \in \mathbb{N}$ with $p$ a prime number and $n \in \mathbb{N}$

Apparently this question requires a method linked with linear algebra but I was wondering if it was possible to solve it in a formal way like an induction on $n$ or by using an identity for $p^{n}-1$ ...
3
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2answers
77 views

How to evaluate $\lim\limits_{n\rightarrow \infty} \sin (1/n)$?

I know that the $\lim\limits_{n\rightarrow \infty} \sin \left( \frac{1}{n} \right)= 0$ But I am not sure of the right workings. My attempt: As $n$ tends to infinity, $\frac{1}{n}$ will tend to $0$. ...
0
votes
2answers
40 views

Show a $W$ is a subspace and find its dimension

Let $W=\{(a,b,c)\}\in \mathbb{R}^{3} : b=a+c \}$. Show that $W$ is a subspace of $\mathbb{R}^{3}$ and find $\dim(W)$. My solution is : Let $u=(a_1,b_1,c_1)$ and $v=(a_2,b_2,c_2)$ $\in W$ Then ...
3
votes
2answers
57 views

Finding flaw in proof

This is one of the problem I have been working on Velleman's How to prove book: Incorrect Theorem. Suppose F and G are families of sets. If ∪F and ∪G are ...
1
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3answers
38 views

Compare inequalities in a proof by induction

I am solving a proof by induction example. But I ended up with my hypothesis $$ a_{n-1} \geq \frac{2^n}{2}+n^2-2n+1 $$ and my inductive step $$ a_{n-1} \geq \frac{2^n}{2}+\frac{n^2}{2}-\frac{n}{2}. ...
2
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1answer
28 views

1:1 between N and P(N) [duplicate]

For each natural number $n\in\mathbb{N_0}$ list all the sets with at most $n$ elements and with maximal element as and including $n$, in numerical order. Now all the elements of $P(N)$ will be listed ...
1
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1answer
57 views

bounded interval is bounded and connected

Can you please tell me if my proof is correct? Definition: Let $X$ be a subset of $\mathbb R$. We say that $X$ is connected iff the following property is true: whenever $x, y$ are elements in ...
0
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1answer
19 views

How do I finish the proof or make this more rigorous? Outer measure

These are supposed to be a one-liner proof, but I can't make it rigorous for some reason. Find $m^*(rE)$ in terms of $m^*(E)$ where $rE = \{rx: x \in E \}$. Prove $m^*(E\cup F) \leq m^*(E) + ...
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3answers
32 views

The only solution of the equation ${72_8!}/{18_2!}=4^x$ is $x=9$

Problem and Definitions If $n_a!:=n(n-a)(n-2a)(n-3a)\ldots(n-ka):n>ka$, how should I go about solving this?: $$\dfrac{72_8!}{18_2!}=4^x$$ Attempt ...
3
votes
1answer
37 views

Proof of Bernoulli like inequality

I found this inequality and I would like to prove it: $$ (1+x)^n \leq 1 + \frac{nx}{(1-(n-1)x)} $$ with with $n>1$ and $-1<x<1/(r-1)$. Does anybody have an idea? Edit: I added the ...
1
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1answer
32 views

Discrete subspace of $\Bbb{R}$ is countable

Show that any discrete subspace of $\Bbb{R}$ with usual topology is countable. Let $U$ be a discrete subspace of $\Bbb{R}$, for each $x\in U$ choose an interval with rational endpoints ...
1
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2answers
115 views

Understanding proof of Peano's existence theorem

I'm studying the proof of Peano's existence theorem on this paper. At page 5 it is said that the problem $$\begin{cases} y(t) = y_0 & \forall t ∈ [t_0, t_0 + c/k] \\ y'(t) = f(t − c/k, y(t − ...
0
votes
0answers
40 views

Is this proof show that the derivative of zeta function has no zeros in the critical strip?

suppose that :( $k \circ \zeta )(s)$= $(\zeta \circ k )(s) \neq 0 $.....$(1)$ ,and suppose that $ k(s)=\zeta(1-s)$ where : $\xi(s) = s(s-1) \pi^{s/2} \Gamma\left(\frac{s}{2}\right) \zeta(s)$. and ...
2
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2answers
46 views

Alternate way to Prove or disprove $6\mid n(n+1)(n+2)$

This is my proof, I'm wondering if I'm correct, and how to do without induction. My Work Basis Step $$\frac{(1)(2)(3)}{6} = 1$$ Inductive Hypothesis Assume that $\dfrac{k(k+1)(k+2)}{6} = d$ where ...
1
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2answers
48 views

$\lim_{n\rightarrow \infty}a_{n+1}=a_n+\frac{1}{a_n}$

Let there be a sequence define by $a_{n+1}=a_n+\frac{1}{a_n}$. first we can see that $a_{n+1}-a_n=\frac{1}{a_n}\geq$ monotonically increasing let assume there is a limit L so ...
0
votes
1answer
53 views

Find The $\lim_{n\rightarrow\infty}\frac{5^{7n}}{3^{{(n+1)}^2}}$

I am trying to use the Squeeze theorem. is is a valid claim? or is it better left and right hand sides expression to select? $$0\leq\frac{5^{7n}}{3^{{(n+1)}^2}}\leq\frac{5^{7n}}{3^{n^2}}$$ $0$ ...
0
votes
1answer
26 views

Find number of all $a \in G $ such that $o(a) =3$

Let $G$ be a group and $|G|= 51$ find number of all $a \in G$ such that $o(a)=3$ My solution : by this theorem : if $|G|=pq$ that $ p ,q$ are prime. If $ q\nmid p-1 $ then $\quad$ $G \cong \Bbb ...
0
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1answer
32 views

Show that $(α^{1/u_{n+1}}-1)^{1/(n+1)}<(α^{1/u_{n}}-1)^{1/n}$

Let $α>2$ be a real number. Let $(u_{n})_{n}$ be an increasing sequence. Then my question is: Show that $$(α^{1/u_{n+1}}-1)^{1/(n+1)}<(α^{1/u_{n}}-1)^{1/n}$$ Add. We have: ...
1
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1answer
34 views

Let A⊆R be a set. Prove that A is bounded if and only if there is some M∈R such that M>0 and that |x|≤M for all x∈A.

I proved the above problem as follows but received feedback that I'm not certain I understand. Can someone help me determine where I went wrong in the proof? Let A⊆R be a set. Suppose M∈R where ...
0
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1answer
52 views

Integral of a complex number showing 1=2?

$$\int z \,dz=\dfrac{z^2}{2}$$ $$z=a+bi\implies \int (a+bi) \,dz=\int z \, dz$$ $$a(a+bi)+bi(a+bi)=(a+bi)^2=\dfrac{z^2}{2}=\dfrac{(a+bi)^2}{2}$$ $$ 2(a+bi)^2=(a+bi)^2$$ Assume $z \neq 0$: $$1=2$$ ...
1
vote
1answer
11 views

Every vertex in a caterpillar graph is adjacent to at most two non-leaf vertices

I am not sure about my proof that goes: Use induction on the number of vertex of caterpillar graph, C. Base case, C with n=1 holds since it is a adjacent to no vertex. So the claim holds. Inductive ...
2
votes
1answer
83 views

Verification of a proof that the difference of two odd integers is not odd

Prove or disprove the difference of two odd integers is odd. Here was my answer: $m = 2s+1$ $n = 2t+1$ $m - n = (2s+1) - (2t+1)$ $= 2s - 2t$ $= 2(s-t)$ I then wrote the following: ...
1
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1answer
20 views

Prove that $f_{n}:[1,+\infty[ \rightarrow \mathbb{R} : x \rightarrow \frac{x^n}{1+x^{n}}e^{-x}$ is increasing

By definition, I want to prove that $f_{n+1}(x)-f_{n}(x)\ge0$ for $x \in [1,+\infty[$ So, we obtain : $e^{-x}(\frac{x^{n+1}-x^{n}}{(1+x^{n+1})(1+x^{n})})$ But for $x \in [1,+\infty[$ we have : ...
2
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1answer
31 views

Proof that any set linearly independent has at most $n$ elements (when the vector space has basis with n elements)

My teacher gave us this proof today, but I don't know if I understood it entirely: Theorem: Suppose $V$ a vector space (finitely generated) over the reals. $$B = \{v_1,\cdots, v_n\}$$ where $B$ is ...
2
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0answers
31 views

proof that the lebesgue measure of a subspace of lower dimension is 0.

I asked this question yesterday Lebesgue measure of a subspace of lower dimension is 0, Matt S answer didn't really convince me and after trying to find a solution without using determinants and the ...
0
votes
1answer
21 views

random variables (X,Y) have the following joint PDF

Let the random variable $(X,Y)$ have the following joint PDF $$ f(x,y) = \left\{ \begin{matrix} 2x^{-(2x+y)}, & x>0, y>0\\ 0, & ...
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2answers
52 views

L'Hôpital's rule in proofs

I was asked to prove that $\lim\limits_{x\to\infty}\frac{x^n}{a^x}=0$ when $n$ is some natural number and $a>1$. However, taking second and third derivatives according to L'Hôpital's rule didn't ...
0
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1answer
38 views

Solve by induction.

How would I show this equation is odd by using the induction hypothesis: $$ g(s) = 3(g(s-1))+(g(s-2))+1 $$ I was thinking that I would prove $g(s)$ is odd by $g(s+1) = 3(g(s)+g(s-1))+1$. How would I ...
1
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1answer
22 views

If (an)→ L, an > 0 for all n ∈ N, and L > 0, then prove that √an → √L .

To be honest, I don't even understand what the question is asking, and have no idea how to answer it. Any guidance would be great. I know convergent/divergent definitions, as well as basic limit laws, ...
0
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1answer
42 views

Solve this logical inference

I have the logic inference: Hypotheses: $A \implies (B \lor C)$ $A \lor (D \land B)$ Conclusion: $D \implies C$ I have these equivalence formations: Hypotheses: $A \lor (D \land B)$ $\lnot D ...
2
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1answer
31 views

Trouble Understanding Proof About Polynomials

In the question I have to prove that: There is no polynomial $P (x) = a_n x^n + a_{n−1}x^{n−1} + · · · + a_0$ with integer coefficients and of degree at least 1 with the property that $P(0), ...
1
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3answers
49 views

Prove that the sequence $\cos(n\pi/3)$ does not converge

EDIT: Using a rigorous formal proof, I need to prove that this sequence does not diverge. I of course understand why it doesn't converge... $n=1$ to infinity of course. So, I have a bit of trouble ...
1
vote
1answer
31 views

Verifying $(x_1,y_1) + (x_2,y_2) = (2x_1 - 2y_1,-x_1+y_1)$ is not vector space

I need to prove that this is not a subspace: $$(x_1,y_1) + (x_2,y_2) = (2x_1 - 2y_1,-x_1+y_1)$$ (and with the multiplication definied in a way that I will not write because the sum already fails) ...
0
votes
0answers
21 views

prove strong induction implies weak induction

So trying to prove: $[s(n_0)\wedge s(n_1)\wedge\cdots \wedge s(n_k)\wedge\forall_n[s(n-k)\wedge s(n-k+1)\wedge\cdots \wedge s(n-1)\wedge s(n)\rightarrow s(n+1)]\Rightarrow \forall_{n_0\le n}s(n)]$ ...
2
votes
0answers
77 views

$\rm span(S_1) + \rm span(S_2) = \rm span(S_1 \cup S_2)$ for infinite sets

I have these two definitions of span: Span: Suppose a vector space $(V,+,\cdot)$, and $$S = \{u_1,\cdots,u_n\}$$ (and $S$ is a subset of $V$, not a subspace) ...
0
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0answers
24 views

Find mean value of amount of Hamiltonian cycles in the random complete directed graph

We are given the random tournament (randomized uniformly) on $n$ vertices. Task is to find the average value of the amount of Hamilton cycles on that tournament. This problem was covered in the ...
1
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2answers
61 views

Can $\mathbb{Z}_{6}$ be a subring of some field?

I think the answer is yes because $\,\mathbb Z_{6}\,$ is a ring, it has a unity and has multiplicative inverse and its elements are commutative so it can be a subring of a field. Is this correct? ...
0
votes
1answer
50 views

A question about the proof that $f_n\to f $ in $L^p$ iff $\lbrace \vert f_n\vert^p\rbrace$ is Uniformly Integrable.

Update: The following question is about a proof of Theorem 8, Section 7.3 of Royden's Real Analysis, 4e. How is the fact that $\lbrace \vert f\vert^p\rbrace$ is integrable over $E$ used in the last ...