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

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5
votes
1answer
166 views
+50

Conditional expectation $\mathbb E\left(\exp\left(\int_0^tX_sdB_s\right) \mid \mathcal F_t^X\right)$

I have found a theorem (see below) in two papers an I try to figure how it could be proved. The result seems to be intuitive, but I'm not able to prove it in a rigorous way. Assumptions: Consider a ...
0
votes
0answers
19 views

finding the supremum

Let $A=\{x:\frac{[b\cdot n]}{n}\}$ when $n\in \mathbb{N}$ find the supremum we know that $b\cdot n-1<[b\cdot n]<b \cdot n$ therefore $b- \frac{1}{n}<\frac{[b\cdot n]}{n}<b$ so b is a ...
0
votes
1answer
23 views

finding infimum

find the infimum and supremum of $E=\{x \in \mathbb{R}:x=\frac{2}{n}+(-1)^n, n\in \mathbb{N}\} $ $Max(E)=2$ therefore it is also the $Sup(E)$ Let assume that there is $-1<m: m\in E$ so $-1$ ...
1
vote
2answers
51 views

Why $ax^2+bx+c = a(x-r)(x-s)$, where $r$, $s$ are the roots?

When I was reading about math, I came across the following - Suppose the roots of the quadratic $ax^2+bx+c$ are $r$ and $s$. Then $ax^2+bx+c = a(x-r)(x-s)$ for all values of $x$. Is there ...
0
votes
1answer
41 views

Biggest n that can be solved in one second

I have been given the following problem: What is the largest n for which one can solve in one second a problem that requires $(\log_2(n))^2$ elementary operations, where each elementary operation is ...
0
votes
0answers
20 views

How is floor(x) + j/A between x and y, and how the proof is related to the well-ordering property?

I am having trouble with a proof by well-ordering property exercise. Use the well-ordering principle to show that if x and y are real numbers with $x \lt y$, then there is a rational number r with $x ...
0
votes
1answer
34 views

Looking for Clarification on a proof of Density of Q in R

I am looking for some advice/help in regard to the proof that Q is dense in R, given in Walter Rudin's book "Principles of Mathematical Analysis". Mostly, I want to see if my reasoning is correct for ...
2
votes
1answer
65 views

Is this simple calculus proof formal enough and correct?

There is function $f$ differentiable at $x=0$ and $f'(0) = m > 0, f(0) = 0.$ I need to prove that there is $K > 0$ and $\delta > 0$ that for every $0<x<\delta$ : $f(x) > Kx$. So I ...
1
vote
1answer
17 views

An affine set $C$ contains every affine combinations of its points

Show that an affine set $C$ contains every affine combinations of its points. Proof by induction: From the definition of an affine set, we know that $\forall x_1,x_2\in C \text{ and } \theta_i\in ...
0
votes
1answer
15 views

If$\mu$ is $\sigma$- finite, $\epsilon>0$, there exists $A\in \mathcal{A}$ such that $\mu(A)<\infty$ and $\epsilon+\int_A f>\int f$

Problem Let $X\mathcal{A},\mu$ be a $\sigma$-finite measure space. Suppose $f$ is non-negative and integrable. Prove that if $\epsilon>0$, there exists $A\in \mathcal{A}$ such that ...
7
votes
2answers
132 views

proof for $\frac{1}{i} = -i$?

My physical chemistry textbook seems to be making the implicit assumption that $\cfrac{1}{i} = -i$. I'm not sure how this is valid. Here is the snippet of relevant steps: ...
4
votes
3answers
71 views

Show that $f$ is continuous mathematically.

Let $f:[0,\infty)\to \mathbb{R}$ be given by $f(x)=\sqrt{x}$. Show that it is continuous. This is taken from Example 3.7 on <link> page 22 on the paper. It has shown that it is continuous at ...
0
votes
1answer
32 views

Quick question about basic proofs (Spivak)

I was taking a look at some of the practice question from the first chapter of Spivak, and I am wanting to just verify if I am on the right track with things. I do not have solutions, so I am just ...
11
votes
2answers
105 views

Proof that $\sqrt{F!-1}$ is irrational

Please tell me whether my proof is valid. (1) Suppose $\sqrt{F!-1}= \frac p q$ where $p, q$ are integers $>0$ with no common factors. (If there are any common factors we cancel them in the ...
2
votes
1answer
45 views

Analysis: Is $A$ dense in $[0,1]$?

Let $f_n:[0,1]\to\mathbb R$ define by $f_n(x)=\cos(nx)$. Let $A_n=\{x\mid f_n(x)=0\}$ and $A=\bigcup_{n\in\mathbb N}A_n$. I have shown that $|A|=+\infty$. Do you think that $A$ is dense in $[0,1]$ ? I ...
0
votes
1answer
35 views

Point-Set Topology Proof Verification

I am self-studying Baby Rudin (after some idleness) and his second chapter is on basic point-set topology. Now, he asks the following: Is every point of every open set $E\subset\mathbb{R}^2$ a limit ...
2
votes
0answers
17 views

Using Erdős–Szekeres theorem for graph with 50 vertices

Let $G$ be a graph with $50$ vertices such that for every $4$ vertices there are $2$ that have no edge between them (independent). I want to prove that $G$ has a group of $5$ independent vertices.My ...
1
vote
0answers
15 views

Quickest way to restrict a homeomorphism

Let $\phi: U \to V \subset \mathbb{R}^n$ homeomorphism. My desire is: I want to say the restriction $\phi|_{\phi^{-1}(B_{r'}(x))}:\phi^{-1}(B_{r'}(x)) \to B_{r'}(x) $ is a homeomorphism in the ...
2
votes
1answer
52 views

Anywhere I integrate $f_n$, the integral approaches $f$. Is $\lim_n f_n = f$ a.e.?

Something tells me this is obvious... I have a bunch of functions: $f,f_n:\mathbb{R}^2\rightarrow \mathbb{R}$, all integrable. Also, $f$ is continuous. I also have a family of sets, $\mathcal{G}$ ...
2
votes
1answer
46 views

Every continuous map of a closed interval into itself has a fixed point

The Question: Please show this theorem: Let $f: I=[a,b] \rightarrow \mathbb{R}$ be a continuous map such that $f(I) \supset I $. Then $f$ has a fixed point on I. My Attempt: Suppose there is a ...
3
votes
0answers
54 views

Homework problem about the smallest sigma algebra:

Let $\mathscr{E} \subset 2^X $, then there is a unique smallest $\sigma$-algebra containing $\mathscr{E} $. Proof(Attempt): Since $\mathscr{E} \subset 2^X $, $2^X$ is $\sigma$-algebra containing ...
5
votes
1answer
109 views

Question about proving basic results of numbers

I have just recently started to work with Calculus by Spivak and I am just wondering some things about the first chapter. ( I am doing this as a method to review my calculus which I have done but only ...
1
vote
1answer
41 views

$f$ is continuous $\iff f(\bar A) \subset \overline{f(A)}$

The problem is: $f:X\to Y$: any map. $f$ is continuous $\iff \forall A\subset X, \ f(\bar A) \subset \overline{f(A)}$ My understanding is: Suppose $f$ is continuous. $\forall A\subset X, A ...
1
vote
1answer
50 views

Is my approach right? Or is there a better way?

Let $E,F$ normed linear spaces, let $C$ connected of $E$, $D\subset F$, and $f:C\to D$ such that $f$ is open (i.e. sends open sets in $C$ "which is the same as open sets of $E$ intersected with ...
3
votes
2answers
94 views

Proof of the derivative of $x^n$

I am proving $(x^n)'=nx^{n-1}$ by the definition of the derivative: \begin{align} (x^n)'&=\lim_{h \to 0} {(x+h)^n-x^n\over h}\\ &=\lim_{h \to 0} {x^n+nx^{n-1}h+{n(n-1)\over ...
1
vote
1answer
40 views

proof of rational numbers as repeating or terminating decimald

As an exercise in my conceptual algebra class we attempted to determine the reason why this theorem holds true in the forward direction. (Note we decided not to tackle the opposite direction) I wrote ...
1
vote
2answers
78 views

Showing that $f$ is $C^\infty$

Question: Let $f: U \to \mathbb R$ be a continuous function, with $U \subset \mathbb R^2$ open, such that $$(x^2 +y^4)f(x,y) + f(x,y)^3 = 1,\, \,\, \forall (x,y) \in U$$ Show that $f$ is of ...
0
votes
1answer
20 views

Is this statement equivalent to $f(x)\in\mathscr C(a,b)$?

I'm pondering on the following: $$f(x)\in\mathscr C(a,b)\overset{?}{\Longleftrightarrow} f(x)\in\mathscr C[a+\delta,b-\delta]\quad\forall\delta\in(0,\frac12(b-a)) $$ I believe it's true. The ...
1
vote
1answer
81 views

About the usage of the strong approximation theorem

I'm reading Henning Stichtenoth's Algebraic Function Fields and Codes and at Proposition 3.2.5(a) of section 2, chapter 3 he says: Let $\mathcal{O}_S$ be a holomorphy ring of $F/K$. Then $F$ is ...
2
votes
1answer
16 views

Show that $\text{ord}_n (ab) \mid \text{ord}_n (a) \cdot \text{ord}_n (b)$ when $(a,n)=(b,n)=1$ and $(\text{ord}_n (a), \text{ord}_n (b)) = 1$.

The question is to show that if $n$ is a positive integer with $(a,n)=(b,n)=1$ and $(\text{ord}_n (a), \text{ord}_n (b)) = 1$, then $\text{ord}_n (ab) \mid \text{ord}_n (a) \cdot \text{ord}_n (b)$. I ...
2
votes
2answers
45 views

Verification for this proof

Sorry guys about the verification questions but it's near the end of the semester and I am very sheepish about making mistakes especially because real analysis is a very important course it's only the ...
0
votes
1answer
33 views

Showing the following collection is a $\sigma$-algebra of sets

Let $X$ be any uncountable set. We show the following $$ \mathscr{A} = \{ E \subset X : E \; \; \text{is countable or} \; \; E^c \; \; \text{is countable} \} $$ Suppose $E \in \mathscr{A} $, then ...
0
votes
0answers
20 views

Can this be a way to prove that the set of all limit points is closed

Let $x\in A'$ where $A'$ is the set of all limit points of $A$ . Let $x$ be a limit point of A'. This implies every neighborhood $x$ contains some p such that $p\neq x$ and $p\in A'$. This implies ...
2
votes
0answers
44 views

Alternative proof for the equality of two angles in an isosceles triangle.

From the answers of my previous question, I got an idea to prove equality of two angles in an isosceles triangle. In that question the equality of two angles in a right-angled-isosceles triangle was ...
1
vote
2answers
46 views

Are these rings fields?

Are the following rings fields? 1) $\Bbb Q[x] /\langle x^2+1\rangle$ Since a polynomial ring taking values on any field is a E.D, and hence a P.I.D, this is a field iff the ideal is prime or ...
1
vote
1answer
21 views

Suppose $\mu$ is a finite measure and $\sup_n \int |f_n|^{1+\epsilon} \ d\mu<\infty$ for some $\epsilon$. Prove that $\{f_n\}$ is uniformly integrable

Problem Suppose $\mu$ is a finite measure and $\sup_n \int |f_n|^{1+\epsilon} \ d\mu<\infty$ for some $\epsilon>0$. Prove that $\{f_n\}$ is uniformly integrable. Background A family $\{f_n\}$ ...
0
votes
2answers
37 views

A question on a proof of $C^1$ implies locally Lipschitz

I stumbled upon this answer here while studying the proposition that if $f: \mathbb R^n \to \mathbb R^n$ is $C^1$ then $f$ is locally Lipschitz. The answer in the link applies Taylor's theorem. And ...
1
vote
3answers
50 views

False proof why $C^1$ implies locally Lipschitz

I produced a (possibly) false proof of why $C^1$ implies locally Lipschitz for $f: \mathbb R^n \to \mathbb R^n$. Please could someone tell me where my mistake is? Proof: Let $f: \mathbb R^n \to ...
0
votes
0answers
27 views

Continuos, surjective map $\pi$ is a quotient map $\iff$ $\pi$ sends saturated open to open or saturated closed to closed

Problem is: Continuos, surjective map $\pi$ is a quotient map $\iff$ $\pi$ sends saturated open to open or saturated closed to closed. ($U$ is saturated $\iff$ $\exists V \in Y$ s.t. $U = \pi^{-1} ...
2
votes
0answers
16 views

$U$ takes the same value on $\pi$ then $U$ is saturated

Let $\pi : X \to Y$ be any map, and $U$ be a subset of $X$. The problem is: "$\forall x\in U$, $ \pi (x) = \pi(x') $, then $x' \in U$" then $U$ is saturated. (U is saturated $\iff$ $\exists V ...
-1
votes
0answers
52 views

Why doesn't -1 = 1 (spot the falacy) [duplicate]

I'm trying to figure out the fallacy in this statement: j = $ \sqrt{-1} = \sqrt{1/-1} = \frac{ \sqrt{1}} {\sqrt{-1}} = \frac 1j $ $ =>j^2=1 $ but $j^2 = -1 $ $ =>-1=1 $ My text book says ...
0
votes
1answer
33 views

Prove that if function f is monotonic, then it one-to-one

What I have so far: Suppose $f$ is monotonic. It is therefore either increasing or decreasing. Proof for increasing: If $f$ is increasing, then $f(x_1) <f(x_2)$ whenever $x_1 < x_2$, which ...
0
votes
0answers
14 views

Continuous map of cadlag functions (one sided limits exist and right continuous) is cadlag

Recall that a real function $f$ is cadlag if the one sided limits $f (t^-), f (t^+)$ exist and $f (t^+) = f (t)$, i.e. $f$ is right continuous. Then is the following true? If $f$ is continuous and ...
0
votes
0answers
46 views

Determine all maximal and prime ideals of the polynomial ring $\Bbb C[x]$

Determine all maximal and prime ideals of the polynomial ring $\Bbb C[x]$ My attempt: Note that $\Bbb F[x]$ where $\Bbb F$ is any field is a Euclidean domain, and importantly, that means that ...
4
votes
3answers
32 views

Is the set of all rational numbers with odd denominators a subring of $\Bbb Q$?

Is the set of all rational numbers with odd denominators a subring of $\Bbb Q$?(When the fraction is completely reduced) I have tried to apply the subring test on this, and this means I want to show ...
0
votes
0answers
34 views

Is it true that $\mathbf{P} (X \leqslant t) =\mathbf{P} (F (X) \leqslant F (t))$ for continuous $X$

For continuous $X$ with distribution $F$, is it true that $\mathbf{P} (X \leqslant t) =\mathbf{P} (F (X) \leqslant F (t))$? Also is continuity required? I've attempted a proof: Since $\mathbf{P} (F ...
0
votes
1answer
44 views

Please could someone check my proof that continuous implies locally Lipschitz

I have produced a false proof but can't spot the mistake. I proved the following (false) statement: Let $U \subseteq \mathbb R^n$ be open and $f: U \to \mathbb R^n$ be continuous. Then $f$ is ...
2
votes
1answer
46 views

Prove $((A^C \cup B^C) \setminus A)^C = A$

I have attempted this proof as outlined below. However, I feel that it is not correct, any suggestions would be appreciated. Prove $((A^C \cup B^C) \setminus A)^C = A$ L.H.S. $((A^C \cup B^C) ...
1
vote
2answers
54 views

Trying to understand a proof about onto/1-1 mappings (from Herstein's Topics in Algebra)

I am working on some problems in a book I have and I want to make sure that I have an accurate possible proof. That is, I want to make sure I actually understand/ can justify the reasoning. (some of ...
3
votes
3answers
49 views

A proof that $EX_n\to EX$ for uniformly integrable $\{X_n\}$ with $X_n\to X$ a.s.

I'm having some trouble following someone's proof of the following result: Assume that $\{X_n\}$ are uniformly integrable and that $X_n\to X$ a.s.; then $EX_n\to EX$. First, the author shows that ...