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

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0answers
23 views

Dubiety About An Inequality Proof

In Principles of Mathematical Analysis, the author is attempting to demonstrate that, if $x>0$ and $y<z$, then $xy<xz$, which essentially states that multiplying by a positive number does not ...
2
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0answers
65 views

Is this a valid proof for $1+1=2$? [duplicate]

I am extremely new to proofs, and quite bad at them. In studying and practicing the different types of proofs, I developed this very rough proof that $1+1=2$, one of the simplest mathematical truths I ...
2
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1answer
43 views

$\dim V = \dim \phi(V)+\dim \ker \phi$

I want to show that $\dim V = \dim \phi(V)+\dim \ker \phi$. I know this proof can be found in any linear algebra textbook. However, my question is not exactly about the proof, but on a statement I ...
4
votes
1answer
84 views

If $f(\mathbb{R})$ is compact and $f$ is continuous, then is $f$ uniformly continuous?

Question: If $f(\mathbb{R})$ is compact and $f$ is continuous, then is $f$ uniformly continuous? Background: I thought of the question when proving that "If a function is periodic and continuous, ...
5
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2answers
260 views

Simple proof that $\pi$ is irrational - using prime factors of denominator

Simple proof that $\pi$ is irrational Consider the Gregory - Leibniz series for $\pi/4$: $$\frac \pi 4 = 1 - \frac 1 3 + \frac 1 5 + \cdots $$ Let $A_n/B_n$ be the irreducible fraction given by ...
2
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1answer
28 views

Proving by Cauchy's definition $\lim_{x\to 0} x^2\cos x=0$

Prove by definition that $$\displaystyle\lim_{x\to 0} x^2\cos x=0$$ So take $\delta=\sqrt\epsilon$, and from definition we have: $|x|<\delta\Rightarrow|x^2|<\delta^2\Rightarrow|x^2\cos ...
4
votes
2answers
233 views

Did I do this proof right?

I am not sure if I did the proof right, so I wanted to see how most of you did this. I am trying to solve this problem: Let $x, y \in \mathbb N$ be relatively prime. If $xy$ is a perfect square, ...
2
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0answers
40 views

Is this proof clear, complete and readable?

I am trying to prove this statement: $C^2/U(1) $ can be identified with $R^3$ so that the image of the $U(1)$ fixed point is $(0,0,0)$. And I was wondering if someone could tell me if the following ...
2
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5answers
71 views

Every sequence in $\mathbb{R}$ has a monotonic subsequence

I have trouble with this kind of infinite construction in topology. Can someone check my proof is sound? Let $s$ be a sequence in $\mathbb{R}$. Then $s$ has a monotonic subsequence. There are two ...
3
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2answers
44 views

$\Bbb{Q}$ is not a finitely generated $\Bbb{Z}$-module

I'm trying to show that $\Bbb{Q}$ is not a finitely generated $\Bbb{Z}$-module. Assume to the contrary that $$\Bbb{Q}=\Bbb{Z}\dfrac{a_1}{b_1}+...+\Bbb{Z}\dfrac{a_n}{b_n}$$ where $a_i,b_i\in\Bbb{Z}$. ...
1
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1answer
32 views

Ordered Field: $|x|\le y$ iff $-y\le x\le y$

I had a question regarding this part of a theorem that describes the inequalities of the absolute value function for order field $\mathbb{F}.$ Here is the theorem: Theorem: Let $\mathbb{F}$ be an ...
1
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3answers
56 views

proof of chain rule

Is my proof correct? show: $(g\circ f)'(x_0)=g'(y_0)f'(x_0)$ Since $f'(x_0) = \lim_{x\to x_0} \frac{f(x)-f(x_0)}{x-x_0}$ and Since $g'(y_0) = \lim_{y\to y_0} \frac{g(y)-g(y_0)}{y-y_0}$ Multiply ...
0
votes
1answer
25 views

differentiabilty implies continuity (analysis)

Is my proof correct? We need to show that if $f$ is differentiable at $x_o$, then it is continuous at $x_o$ i. e. $$\forall \epsilon >0, \exists \delta >0 \text{ s.t. } ...
0
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1answer
42 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
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1answer
46 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 ...
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4answers
57 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
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1answer
42 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
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1answer
29 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
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1answer
67 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
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1answer
42 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
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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
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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
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0answers
67 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
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1answer
70 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
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1answer
78 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
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3answers
43 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
48 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
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2answers
63 views

About the infinitude of some kind of primes? [closed]

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
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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
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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
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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
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1answer
28 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, ...
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1answer
46 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
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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
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1answer
27 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
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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
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0answers
33 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
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0answers
28 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) ...
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1answer
35 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 $I$. ...
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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 ...
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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
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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
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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
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2answers
141 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
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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
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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 ...