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

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1answer
36 views

Inverse functions and derivatives

Suppose that $f:[a.b]\to[c,d]$ is differentiable and onto. If $f'$ is never 0 on $[a,b]$ and $d-c\geq2$, prove that for every $x\in[c,d]$, there exist $x_1\in[a,b]$ and $x_2\in[c,d]$ such that $|f'(...
1
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1answer
32 views

Solving $\exp\bigg(\frac{2+\pi i}{4} \bigg)$

$\exp\bigg(\dfrac{2+\pi i}{4} \bigg)$ My try: $=e^{(1/2)}e^{(\pi i)/4}$ $=e^{(1/2)}[\cos(\pi/4)+i\sin(\pi/4)]$ My try is correct?
2
votes
2answers
48 views

Prove that $a$ and $b$ are coprime whenever $a+b$ and $a-b$ are coprime

I wish to prove that if $a+b$ and $a-b$ are coprime, then $a$ and $b$ are coprime. I want to make sure that my proof holds, so any reasonable insight concerning this problem would be appreciated. ...
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0answers
27 views

Hölder's inequality using Lagrange multiplications. Where am I wrong?

The specific question is a series of question in the end of which I am to show the correctness of Hölder's inequality, and therefore I need some guidance. Show that for $f(x,y)=a_1x+a_2y$ under the ...
2
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1answer
55 views

How to prove equality with exponential function

How to prove this equality: $$\frac{1-e^{-\frac{1}{t}}}{1-e^{-\frac{1}{2t}}}=1+e^{-\frac{1}{2t}}$$ I've no idea how to start so everything is welcome! Thanks a lot!
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0answers
36 views

Interchange intersection and union in proof of Blumenthal’s zero-one law

I am trying to prove Blumenthal's zero-one law using Kolmogorov's zero-one law. I use that $B_t$ Brownian $\iff$ $tB_{1/t}$ Brownian. Can I change the intersection and union as follows? Start of my ...
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3answers
78 views

Alternative proofs (algebra)

I wrote some short alternative proofs (sketches mostly) to my book, can someone tell me if they are okay. The unity of a subfield is the unity of the whole field. Let $H \subset F$ with $F$ ...
5
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1answer
97 views

Winning Strategy with Addition to X=0

Problem: Two players play the following game. Initially, X=0. The players take turns adding any number between 1 and 10 (inclusive) to X. The game ends when X reaches 100. The player who reaches 100 ...
1
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2answers
86 views

What is the flaw in this induction proof? [duplicate]

Explain the flaw in the following induction argument which shows all of Lucas’ toys are the same colour. Proof: We will show by induction that: for every integer $n\ge1$, in any group of $n$ of Lucas’...
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1answer
89 views

Prove $x_n$ converges IFF $x_n$ is bounded and has at most one limit point

I'm not entirely sure how to go about proving this so hopefully someone can point me in the right direction. The definition I have for a limit point is "$a$ will be a limit point if for a sequence $...
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0answers
38 views

Prove that the function $f:[2,4]\rightarrow\mathbb{R}$ is integrable.

Define $$f(x)=\begin{cases}x\qquad\text{if $2\leq x\leq 3$}\\ 2\qquad\text{if $3<x\leq4$}\end{cases}$$ Prove that the function $f:[2,4]\rightarrow\mathbb{R}$ is integrable. Let $f=f_1+f_2$ ...
1
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1answer
29 views

“limit-truncation” argument in Donsker's theorem

I'm working out the proof of Donsker's theorem given in Revuz and Yor, Continuous martingales and BM, (theorem 1.9, ch. XIII, p. 518). At one certain point, they prove the following fact : for every ...
0
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1answer
39 views

Prove that $\mathscr{G}$ is Countable

Let $\mathscr{G} = \{N(p;r) \, : \, \, p, r \in \mathbb{Q} \, \, \, \mathrm{and} \, \, \,r > 0\}$. (a) Prove that $\mathrm{G}$ is countable. (b) Let $A$ be a nonempty open set and let $\mathrm{G}...
0
votes
1answer
87 views

multi choose proof problem

I am having trouble creating a proof for the below equation to show that they are in fact equal to one another. n and k are positive integers. $$ \left(\!\!{n\choose k}\!\!\right) = \sum_{i=0}^{k}\ \...
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1answer
29 views

Verification of my solution to $\frac{dy}{dx}=\frac{2xy^2+x}{x^2y+y}$

I'm not certain that the following solution is entirely correct: $$\frac{dy}{dx}=\frac{2xy^2+x}{x^2y+y}=\frac{1}{y}\frac{x(2y^2+1)}{1+x^2}$$ $$\frac{y\frac{dy}{dx}}{2y^2+1}=\frac{x}{1+x^2}$$ $$\frac{...
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2answers
84 views

How many elements of order 5 might be contained in a group of order 20?

Exercise from Artin's 2nd edition of Algebra. How many elements of order $5$ might be contained in a group of order $20$? My attempt: By the third Sylow Theorem, $|Syl_{5} (G)| = 1, 2, 4$ and it is ...
0
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0answers
13 views

Relation deterministic and stochastic little-0 notation

I have a question on the relation between $o(.)$ and stochastic $o_p(.)$. In van der Vaart Lemma 2.12 p.13 we read that Let $R$ be a function defined on the domain $\mathbb{R}^k$ such that $R(0)=...
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1answer
34 views

Determining the exact one from all possible Jordan Canonical Forms of a matrix

Here is the example I encountered : A matrix $\ M\ $ $(5\times 5)$ is given and its minimal polynomial is determined to be $(x-2)^3.$ So considering the two possible sets of elementary ...
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2answers
52 views

Show that for any Sets $A$ and $B$, $\mathcal{P}(A) \cap \mathcal{P}(B) = \mathcal{P}(A\cap B)$

Question: show that for any sets $A$ and $B$, $\mathcal{P}(A) \cap \mathcal{P}(B) = \mathcal{P}(A \cap B).$ I want to prove it. Consider the following attempted proof. $$(1)\:\:\:A \cap B \in \...
6
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1answer
46 views

Prove H is in the center Z(G)

Exercise from Artin's 2nd edition of Algebra. Let $H$ be a normal subroup of prime order $p$ in a finite group $G$. Suppose that $p$ is the smallest prime that divides the order of $G$. Prove that $H$...
3
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4answers
208 views

Proof that the number $\sqrt[3]{2}$ is irrational using Fermat's Last Theorem

Suppose that $\sqrt[3]{2}=\frac{p}{q}$. Then $2q^3 = p^3$ i.e $q^3 + q^3 = p^3$, which is contradiction with Fermat's Last Theorem. My question is whether this argument is a correct mathematical ...
2
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1answer
167 views

Proof of “Singular values of a normal matrix are the absolute values of its eigenvalues”

I want a simple proof of this fact using only definitions and basic facts. I've searched for it for some time and I couldn't find a satisfying proof. So I attempted to do it myself. Let $A \in \...
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0answers
22 views

Question involving multivariable calculus.

The temperature in a neighbourhood of the origin is given by a function $$T(x,y) = T_0+e^y \sin x$$ A heat fleeing particle is placed at the origin at time t = 0. find the differential equations on ...
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1answer
37 views

Prove that the sequence given by $x_1=1, x_{n+1}=x_n+\frac{1}{x_n^2}$ is unbounded [closed]

Prove that the sequence given by $x_1=1, x_{n+1}=x_n+\frac{1}{x_n^2}$ is unbounded. It is enough to prove that $\lim_{n\rightarrow\infty} x_n = \infty$. Any hint please?
0
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1answer
29 views

Show that any closed ball in $E$ is entirely contained in at least one set $U_i$

Assume that for every $\epsilon>0$ there is a closed ball in $E$ of radius $\epsilon$ that is not contained in any of the sets $U_i$. As suggested, consider the sequence of closed balls $\bar B(p_1,...
1
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1answer
26 views

Prove that $\lim_{n\to\infty} \sup{x_{n}}=\lim_{n\to\infty} \inf{x_{n}}=L$ IFF the sequence $(x_n)$ converges.

Prove that $\lim_{n\to\infty} \sup{x_{n}}=\lim_{n\to\infty} \inf{x_{n}}=L$ IFF the sequence $(x_n)$ converges. My attempt: $\Rightarrow$ Since $\inf{x_n}\leq x_{n} \hspace{1mm} \forall n\in\mathbb{N}$...
2
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1answer
58 views

Prove that the sequence $a_1, a, a_2, a, a_3, a,\ldots$ converges iff $a_1,a_2,a_3,\ldots$ converges

Prove that the sequence $a_1, g, a_2, g, a_3, g,\ldots$ converges to $g$ iff $a_1,a_2,a_3,\ldots$ converges to $g$. Obviously, if $a_1, g, a_2, g, a_3, g,\ldots$ converges to $g$, then its ...
3
votes
1answer
53 views

If $f$ is one-to-one and continuous on the closed interval $[a,b]$ then prove that $f$ is strictly monotone on $[a,b]$

If $f$ is one-to-one and continuous on the closed interval $[a,b]$ then prove that $f$ is strictly monotone on $[a,b]$. So my plan was to prove this by contradiction. I'm wondering if there is a ...
0
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1answer
51 views

Weak convergence implies uniform convergence

I'm trying to show that pointwise convergence in distribution implies uniform convergence in distribution when the limiting cdf is continuous everywhere using the proof in van der Vaart "Asymptotic ...
3
votes
0answers
60 views

Is there any mistake in my proof?

My little brother started fiddling around with his calculator, and noticed something curious: $$ \Large \sqrt{a\cdot\sqrt{a\cdot\sqrt{a\cdot\sqrt{a \cdot \ldots}}}} = a $$ So I went ahead and wrote a ...
3
votes
4answers
109 views

Is this proof of $(xy)^{-1}=y^{-1}x^{-1}$ valid?

Proof. Let $G$ be a group, and $x,y\in G$. Then, we have $xy(xy)^{-1}=e$, where $e$ is the identity element of $G$. Multiplying on the left by $x^{-1}$, we get $y(xy)^{-1}=x^{-1}$. Multiplying once ...
2
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1answer
63 views

Prove or disprove: For every integer $k\in \mathbb{Z}$, if $f(x)$ is additive, then $f(kz)=kf(z)$.

Given the definition of additive: A function $f:\mathbb{R} \to \mathbb{R}$ is called additive if $f(x+y)=f(x)+f(y)$ for all $x,y\in \mathbb{R}$. We want to prove or disprove the following claim: ...
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5answers
150 views

Prove number of handshakes between $n$ people is $\tfrac{n(n−1)}{2}$ by induction [closed]

How do we calculate the number of handshakes between $n$ people? And where do I apply the inductive step?
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1answer
88 views

Find number of ring homomorphisms from $\mathbb Z_{12} \to \mathbb Z_{28}$ [duplicate]

Find number of ring homomorphisms from $\mathbb Z_{12} \to \mathbb Z_{28}$. This question has been asked before but I found the solution confusing.Please check whether my approach is valid. A ring ...
1
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1answer
36 views

Area under the convolution proof

Prove that $\int\limits_{-\infty}^\infty (f*g)(x)dx=\left(\int\limits_{-\infty}^\infty f(x)dx\right)\left(\int\limits_{-\infty}^\infty g(x)dx\right)$ My proof is: Let $f$ and $g$ be probability ...
2
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2answers
63 views

Non-integral-over-a-point proof that the probability of any point in a continuous distribution is zero

My Question For continuous random variables / continuous distributions, it is defined that the probability of any point has probability $0$. The most common proof for this is as follows: $$\Pr(X=a)=\...
2
votes
2answers
78 views

Integrate $\int \frac {\ln x}{x^2}$ by U-Substitution

I've got a question regarding "improper" u-substitution integrals. How would we solve $\int \frac {\ln(x)}{x^2}dx$ using the u-substitution $u = \ln(x)$? I know that it might be easier to solve it ...
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2answers
44 views

Mistake in a proof concerning centers of incircles

In the above diagram we have that $D, E, F$ are the points of contact of the incircle of an acute-angled triangle $\Delta ABC$ with $BC, CA, AB$ respectively ,and $I_1,I_2,I_3$ are the incentres of ...
2
votes
1answer
90 views

Proof that product topology of subspace is same as induced product topology

Let's assume that $A\subseteq X$ is product of $A_{i}\subseteq X_{i}(i\in I)$. Then product topology of $A$ is the same than the topology induced by $X$. I have proved this few different times now, ...
1
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1answer
60 views

Putnam 2009 A4 clarification

Can anyone verify if this proof of the 2009 A4 Putnam problem is correct? Thanks! 2009 A4. Let $S$ be a set of rational numbers such that I. $0 \in S$. II. If $x \in S$, then $x \pm 1 \in S$. III. ...
3
votes
1answer
53 views

Example that in general $|abc|\neq |cba|$

I tried to provide an example of group elements $a,b,c$ that shows that in general $|abc|\neq |cba|$. Please could you tell me if my example is correct? Let $G$ be the dihedral group $D_6$ (the ...
1
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4answers
115 views

prove that $n(n+1)$ is even using induction

the base case of $n=1$ gives us $2$ which is even. assuming $n=k$ is true, $n=(k+1)$ gives us $ k^2 +2k +k +2$ while $k(k+1) + (k+1)$ gives us $k^2+2k+1$ whats is the next step to prove this by ...
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0answers
17 views

Definition of a vector of rational coordinates

What is a vector of rational coordinates? Consider a random vector $X$ of dimension $k\times 1$ taking values in $\mathbb{R}^k$. Then take $\mathbb{Q}_k:=(q_1,q_2,...)$ as the vectors with rational ...
4
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0answers
61 views

Show that if $\int_I f=0$ for all interval then $f=0$

Let $f$ integrable on $\mathbb R$ and continuous. I have to show that if for all interval $I\subset \mathbb R$, $$\int_If=0$$ then $f=0$. My attempts Suppose $f\neq 0$. Then, there is a $c\in\mathbb ...
2
votes
1answer
57 views

Any open cover of $S^1$ is an open cover of the annulas

The question goes like this : If $\{U_i:i\in I\}$ is an open cover of the unit circle in $\mathbb R^2$ then show that it is an open cover of an annulus $1-\delta\lt ||(x,y)||\...
1
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1answer
51 views

The countable dense subset of every compact metric space

Show that any compact metric space has a countable dense subset. I am having problem with finishing the proof after a few steps. This is how I am going : So, let $X$ be the compact ...
1
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1answer
86 views

Some doubts about easy computations involving nontrivial zeros of Riemann's zeta function

On assumption of Riemann hypothesis when I write a complex zero (nontrivial zero) of zeta function as $\rho=\frac{1}{2}+it_\rho$, and I write $x^\rho$ as $\sqrt{x}e^{it_\rho \log x}$, then multiplying ...
0
votes
0answers
65 views

If $f$ is continuous and positive and $\int_a^b f(x)dx=0$ therefore $f(x)=0$ for all $x\in[a,b]$. [duplicate]

If $f$ is continuous and non-negative and $\int_a^b f(x)dx=0$ therefore $f(x)=0$ for all $x\in[a,b]$. My attempts Suppose $f\neq 0$, i.e. there is an $x\in ]a,b[$ s.t. $f(x)\neq 0$. Since $f$ is ...
0
votes
1answer
74 views

Prove that the set of all binary sequences is uncountable

Question: Prove that the set of all infinite binary sequences is uncountable. Comments: I think that there are a couple of ways of going about this. My first approach was to show that the set of all ...
0
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0answers
47 views

Recall that in the context of Fenchel–Rockafellar duality, the primal problem is defined by

Let X and Y be two Hilbert spaces. Let $f : X \rightarrow ]-\infty, +\infty]$ be convex and lower semicontinuous, let $A : X \rightarrow Y$ be linear, and let $g: Y \rightarrow ]-\infty,+\infty]$ be ...