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

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0
votes
1answer
31 views

Check my logical argument for this proof.

if x is a real number $x \not =\ 1 $, then there exists y which is also a real number $ ((y+1) \div ( y-2) ) = x .$ Prove it's converse also. Logical Argument: given: $x \not = 1$ Goal: $ ...
0
votes
1answer
22 views

Let $(X,\mathfrak T)$ be a topological space and suppose that A and B are subsets of X such that $A \subseteq B$ then $Cl(A) \subseteq Cl(B)$.

Let $(X,\mathfrak T)$ be a topological space and suppose that A and B are subsets of X such that $A \subseteq B$ then $Cl(A) \subseteq Cl(B)$. My definition of closure is "Let $(X, \mathfrak T)$ be ...
1
vote
0answers
29 views

Let $X=\mathbb Z^+$. Topologize $X$ with the finite complement topology. Let $A=\{3,5,6,7\}$. Find $\overline A$, int$(A)$, and bdry$(A)$.

Let $X=\mathbb Z^+$. Topologize $X$ with the finite complement topology. Let $A=\{3,5,6,7\}$. Find closure of $A$ $(\overline A)$, interior of $A$ (int$(A)$), and boundary of $A$ (bdry$(A)$). $A$ ...
3
votes
2answers
37 views

Set theory (containing Power Set) Need Help in a proof

I am confirming whether my proof is correct or not and need help. If $ A \subseteq 2^A , $ then $ 2^A \subseteq 2^{2^A} $ Proof: Given: $ \forall x ($ $ x\in A \rightarrow \exists S $ where $ ...
1
vote
1answer
31 views

If $(x_n)$ is a convergent sequence, for any $\epsilon>0, \exists M$ such that $|x_n-y_n|<\epsilon$ for all $n \geq M$. Is $(y_n)$ convergent?

If $(x_n)$ is a convergent sequence and $(y_n)$ is such that for any $\epsilon>0, \exists M$ such that $|x_n-y_n|<\epsilon, \forall n \geq M$. Is $(y_n)$ convergent? My attempt: Let lim ...
0
votes
1answer
23 views

Give example of convergent and divergent sequences of positive numbers such that lim $x_n^{\frac1n}=1$

So basically I have to give examples where the root test doesn't work, right? For the convergent sequence, I'm taking $x_n=(1,1,1,\cdots)$, then lim $x_n^{\frac1n}=1$ Is there any non-constant ...
2
votes
2answers
34 views

Give an example to show that convergence of $|x_n|$ does not imply the convergence of $x_n$

I'm taking $x_n=(-1)^n=(-1,1,-1,1,\cdots)$, which is divergent, but $|x_n|=(1,1,1,1,1,\cdots)$ converges to $1$. Is this example correct?
0
votes
0answers
7 views

If a set A is bounded from above, then the set of upper bouds M has minimum

I hope the title is clear, because I am italian and I study calculus exclusively from italian books. I had to prove this proposition refusing to look the book (because if I read the proof, I'll ...
1
vote
1answer
13 views

Question on the argument proving primary decomposition theorem

Lang - Algebra p.150, Lemma 7.6 Let $E$ be a torsion module of exponent $p^r(r\geq 1)$ for some prime element $p$. Let $x_1\in E$ be an element of period $p^r$. Let $\bar E = E/(x_1)$. Let ...
0
votes
1answer
18 views

Under What Conditions Is $f:M\rightarrow \mathbb{C}$ Where M Is the Set of 2x2 Matrices a Function and Not a Function?

I came across a problem that I thought was interesting. I attempted to solve the problem below, and I would be grateful if someone would check my logic in what follows. Let the set M of all 2 by 2 ...
0
votes
0answers
25 views

In the field $GF(p^n)$, prove that for every positive divisor d of n, $x^{p^n }-x$ has an irreducible factor over $GF(p)$ of degree d

In the field $GF(p^n)$, prove that for every positive divisor d of n, $x^{p^n }-x$ has an irreducible factor over $GF(p)$ of degree d Let d be any divisor of n. Then $GF(p^d)$ is a subfield of ...
1
vote
2answers
45 views

If a group G is isomorphic to H, prove that Aut(G) is isomorphic to Aut(H)

If a group G is isomorphic to H, prove that Aut(G) is isomorphic to Aut(H) Properties of Isomorphisms acting on groups: Suppose that $\phi$ is an isomorphism from a group G onto a group H, then: ...
3
votes
2answers
35 views

How to prove this Dirac Delta Function property?

How to prove the equation below, using Dirac Delta function properties? $$ \delta(x^2-m^2)=\frac{1}{2|w|}(\delta(x-w)+\delta(x+w)) $$ where $$ w^2=|x|^2+m^2 $$ I tried to show it using $$ ...
-2
votes
1answer
37 views

The Zeros Localization Theorem and the Extreme value Theorem

My intro to analysis book calls the following theorem The Zeros localization theorem: Theorem: Let $p(x)=x^n + a_{n-1}x^{n-1} +\dots+a_1 +a_0,\ x \in \mathbb R$ , be a polynomial. Then all the zeros ...
1
vote
1answer
30 views

Describe explicitly the linear transformation T from $F^2$ to $F^2$ such that $T\epsilon_1=(a,b),T\epsilon_2=(c,d)$

I have to describe explicitly the linear transformation T from $F^2$ to $F^2$ such that $T\epsilon_1=(a,b),T\epsilon_2=(c,d)$ My try: We know that $T\epsilon_1=(a,b),T\epsilon_2=(c,d)$ so let ...
2
votes
2answers
55 views

Proof verification: ab = gcd(a,b)lcm(a,b) without use of prime factorization

I am trying to proof $ab = gcd(a,b)lcm(a,b)$. The definition of lcm(a,b) is as follows: t is the lowest common multiple of a and b if it satisfies the following: i)a | t and b | t ii)If a | c and ...
4
votes
1answer
32 views

real analysis question on equicontinuity

Can anyone verify my proof of the following problem found in Rudin's Principles of Mathematical Analysis chapter 7 exercise 15. Suppose $f$ is a real continuous function on $\mathbb{R}$, ...
0
votes
0answers
29 views

Find the upper sum and lower sum for the following function with respect to the given partition

I have the following homework problem: Find the upper sum and lower sum for the following function with respect to the given partition: Let $s:[0,1]\rightarrow \Bbb R$ be defined by: ...
0
votes
1answer
25 views

Proof Check: Alternating Series Approximation Theorem

Problem: Let $S=\sum\limits_{n=1}^\infty a_n$ be an alternating series where $\vert a_{n+1} \vert < \vert a_n \vert$ and $\displaystyle\lim_{n \rightarrow \infty}a_n=0$. Let $S_n$ be the $n$th ...
5
votes
4answers
60 views

Different Law of Cosines using Sine instead: $c^2 = a^2 + b^2 - 2ab\sqrt{1-sin^2(\theta)}$

Playing around with Trig and the Law of Cosines (LoC), I came up with this formula given a triangle with sides $a$, $b$, $c$ where we are given $a$, $b$ and angle $\theta$ between them: $$c^2 = a^2 + ...
0
votes
0answers
30 views

Prove that for every $n \in \mathbb N$, $\int_{-\pi}^{\pi}f(x)\sin(2nx)dx=0$ if $f(x)$ is odd.

If $f:\mathbb R \to \mathbb R $ is odd continuous function such that $g(x):=f(x + \frac{\pi}{2})$ is even, prove that for every $n \in \mathbb N$, $\int_{-\pi}^{\pi}f(x)\sin(2nx)dx=0$. Since ...
0
votes
2answers
26 views

Should the order of $a^k$ be $h/k$ as opposed to $h/(h,k)$?

Previously shown: Let $m\in\mathbb{N}$ and $a\in\mathbb{Z}$ s.t. $(a,m)=1$. Then the order $d$ of $a$ modulo $m$ exists and $d\mid\phi(m)$. Moreover, whenever $a^k\equiv 1\pmod{m}$, one has $d\mid ...
0
votes
1answer
35 views

Is the following proof rigorous? How could I improve or correct it?

Proof that $\sqrt{2}$ is irrational using the unique prime factorization theorem. My proof: Assume for the purpose of contradiction that $\sqrt{2}$ is rational. By the unique prime fact. the., we ...
2
votes
1answer
20 views

Prove that $A\subseteq B$ if and only if $A^{C}\cup B=\mathscr{U}$

Prove that $A\subseteq B$ if and only if $A^{C}\cup B=\mathscr{U}$. I know we have to show that: if $A\subseteq B$ then $A^{C}\cup B=\mathscr{U}$ if $A^{C}\cup B=\mathscr{U}$ then $A\subseteq ...
1
vote
1answer
54 views

$(B_X,w)$ metrizable implies $X^\ast$ separable

Let X be a normed space and assume that $(B_X,w)$ is metrizable, i.e. the weak topology is metrizable. Show that $X^\ast$ is separable. My attempt: Let $d$ a equivalent metric on $B_X$. For fixed ...
3
votes
2answers
59 views

If $\lim\limits_{z \to \infty} p(z) = \infty$, then $p(z)$ is a constant

Claim: If $p$ is an entire function and $\lim\limits_{z \to \infty} p(z) = \infty$ and $p(z) \neq 0$ $\forall z \in \Bbb C$, then $p(z)$ is a constant. Proof: Define $f(z) = \frac{1}{p(z)}$ so ...
3
votes
3answers
116 views

Proof verification for proving $\forall n \ge 2, 1 + \frac1{2^2} + \frac1{3^2} + \cdots + \frac1{n^2} < 2 − \frac1n$ by induction

Prove by mathematical induction: $\forall n \ge 2, 1 + \frac1{2^2} + \frac1{3^2} + \cdots + \frac1{n^2} < 2 − \frac1n$ Basis Step: (We want to show, $P(2)$, which is 1 + ...
2
votes
1answer
24 views

A clarification of addition on elliptic curves over the complex numbers

I am trying to prove that the order of the two points $P_{\pm}=(0,\pm\sqrt{-g_3})$ is three on the elliptic curve $y^2=4x^3-g_3$, for $g_3 \not= 0$, defined over $\mathbb{P}^2_{\mathbb{C}}$. Here's an ...
2
votes
2answers
40 views

Set Theory Proof: Valid or not?

I'm trying to gain understanding of set proofs and I came across this one. I can't help but think the proof is too simple and that there is more to it. Problem: Prove or disprove for arbitrary sets ...
1
vote
1answer
41 views

Show that if $p$ is an odd prime, then the congruence $x^2\equiv1\pmod{p^{\alpha}}$ has only two solutions, $x\equiv1,x\equiv-1\pmod{p^{\alpha}}$.

Show that if $p$ is an odd prime, then the congruence $x^2 \equiv 1 \pmod{p^{\alpha}}$ has only two solutions, which are $x \equiv 1, x \equiv -1 \pmod{p^{\alpha}}$. Clearly $x \equiv 1, x \equiv ...
2
votes
2answers
46 views

Limit proof check, show $f$ is bounded in a neighborhood of its limit point

edit: as lem has pointed out, the case where x=c is not handled. Could someone suggest an idea? Prove that if a function $f : A \to \mathbb{R} $ has a limit $l \in \mathbb{R} $ at $c \in L(A)$, then ...
2
votes
3answers
121 views

Showing that $f_0 (x_1, \ldots, x_m) \mathrm tr A =\displaystyle{ \sum_{i=1}^n} f_0(x_1, \ldots, Ax_i,\ldots, x_m)$

Question: Consider $f: (-\epsilon, \epsilon) \to \mathbb R^{m^2}$ a differentiable path of matrices $m \times m$ such that $f(0) = I_m$ and the function $g: I \to \mathbb R$ is defined by $$g(t) = ...
2
votes
6answers
85 views

Formally proving that if $x^2 + 1$ is even, then $x$ is odd.

Theorem: If $x^2 + 1$ is even, then $x$ is odd. I have to mention, that I am a beginner at this. So, sorry if it is very wrong. Suppose that $x^2+1$ is even, such that there exists an ...
0
votes
1answer
21 views

Proof that any finite group can be generated by a set of representatives of the conjugacy classes

I have to proof that any finite group can be generated by a set of representatives of the conjugacy classes, but I still had some questions about my proof: Why is $G$ equal to $\bigcup_{g\in ...
4
votes
2answers
132 views

Using Zorn's lemma to show that every field has an algebraic closure.

You may have seen that I posted this proof with some questions earlier today. But I found the answer to most of them. Now I have just one question regarding this proof, so I thought it would be better ...
0
votes
1answer
32 views

A multivariate function with bounded partial derivatives is Lipschitz

I'm curious if I've done this correctly -- please offer suggestions/corrections if not! I'm new to working in $\Bbb R^n$ so clear insights would be appreciated. The problem: Let $f:\Bbb R^2 \to ...
5
votes
1answer
47 views

converge/divergence of a series with a n-th root

for the following series, prove for convergence/divergence $$\sum\limits_{v=1}^\infty \frac{\sqrt[n]{n}}{\sqrt[n]{n!}}$$ $$\sum\limits_{v=1}^\infty ...
0
votes
1answer
30 views

Let $M=\{v∈V,T(v)=d\}$. Prove that $M=v_1+\operatorname{ker}(T)$

Let $T:V\to W$ be a linear transformation. And let d \in Range(T). Hence, $T(v_1)=d$ for some $v_1 \in V$. Now Let $M= \{v\in V | T(v)=d\}$. Prove that $ M = v_1+ker(T)$. i.e. $M = \{v_1+a | a \in ...
0
votes
0answers
24 views

Prove $K\cap L$ is a subspace of V, but $K\cup L$ is never a subspace.

assume K, L are proper subspaces. Prove $K\cap L$ is a subspace of V, but $K\cup L$ is never a subspace. Solution: if $v_1,v_2\in K$, then $c_1v_1+c_2v_2 \ in K$ [because K is a subspace] if ...
1
vote
1answer
32 views

Contradiction in proof that in an integral domain, every prime is irreducible.

Let $\pi$ be a prime element in an integral domain. So, $\pi$ is a non-unit and if $\pi \mid ab \ $ then $\pi \mid a$ or $\pi \mid b$. An irreducible element $z$ is an element such that if $z=ab$, ...
2
votes
3answers
110 views

For any element $g$ of $G,$ where $g$ has order $2,$ define $gH=\{gh│h∈H\}$. Prove that the set $K=H∪gH$ is a subgroup of $G.$

Does this solution make sense? Let $G$ be an abelian group and $H$ a subgroup. For any element $g$ of $G,$ where $g$ has order $2$, define $gH=\{gh│h∈H\}$. Prove that the set $K=H∪gH$ is a ...
6
votes
8answers
200 views

Why is $n^2+4$ never divisible by $3$?

Can somebody please explain why $n^2+4$ is never divisible by $3$? I know there is an example with $n^2+1$, however a $4$ can be broken down to $3+1$, and factor out a three, which would be divisible ...
0
votes
2answers
43 views

Application of Monotone Convergence Theorem

Suppose $f ∈ L^{1}([0, 1])$. Prove that $lim$ $ε→0^{+} \int_{[0,ε]} f dµ = 0$ My attempt at proof: Let $B_N$ be an open ball of radius $N$ centred at origin. $E_N:=$ {$x: f(x)\leq N$} ...
1
vote
1answer
30 views

Limit Of Function Composition

Let there be $g(x)$ and $f(x)$ two functions define on $\mathbb{R}$. And $$lim_{x \to a} f(x)=b$$ $$lim_{y \to b} g(y)=c$$ moreover, $$\ h(y) := \left\{ \begin{array}{l l} g(y) ...
4
votes
1answer
216 views

Prove that there exists an $n\in\mathbb{Z}\cup\left\{-\infty,+\infty\right\}$ such that… (Dynamics)

Let $X=\left\{0,1,2\right\}^{\mathbb{Z}}$ and on it the following dynamics described by $T\colon X\to X$ as follows: A 1 becomes a 2, a 2 becomes a 0 and a 0 becomes a 1 if at least one of its two ...
15
votes
0answers
210 views

Proof of $\zeta(2)=\frac{\pi^2}{6}$

While messing around with some integrals, I have found the following proof for $\zeta(2)=\frac{\pi^2}{6}$, but I'm not sure if it is valid: We take a look at the integral $I=\int_0^{\frac{\pi}{2}} ...
4
votes
1answer
55 views

Let $l$ be a natural number. Prove that $n\lt\sqrt{n ^ 2 + l}\lt n+1$ for almost every $n$.

In my assignment I have to prove the following statement: Let $l$ be a natural number. Prove that for almost every $n$ the following inequality is true: $$n\lt\sqrt{n ^ 2 + l}\lt n+1$$ I chose ...
3
votes
1answer
75 views

Understanding proof of fundamental theorem of algebra

So this is the proof I have: If $p(z)$ is a non-constant polynomial, then there exists a $z \in \Bbb Z$ such that $p(z) = 0$. Let $p(z) = z^n + a_{n-1}z^{n-1} +a_{n-2} z^{n-2} + ... + a_0$ ...
4
votes
3answers
80 views

Proving if $x^3$ is even, then $x$ is even.

Theorem: If $x$ is a positive integer and $x^3$ is even, then $x$ is even. My Proof by Contrapositive: I. Assuming that $x$ is odd, then I will show that $x^3$ is odd. II. $x$ is odd, so $x$ ...
0
votes
0answers
9 views

If solution of NLS is smooth and decaying at infinity; how to justify it satisfy the conservation law?

We consider the cubic nonlinear Shr\"odinger equation(NLS) $iu_{t}+\frac{1}{2} \Delta u = |u|^{2}u, \ u(x, 0)= u_0(x), (x\in \mathbb R^{d}, t\in \mathbb R)$ I have been trying to understand the ...