This tag is for readers who ask for explanation and clarification of some steps of a particular proof.

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4 views

Differentiation under integral sign proof question (special case)

The theorem in my textbook says: Suppose $f(s,x)$ and $f'_s(s,x)$ is continuous on $\alpha < s < \beta$, $a \leq x \leq b$. Then the function $$ \varphi(s) = \int_a^b f(s,x)\,dx $$ is ...
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0answers
2 views

Computing Hilbert transform and envelope of a function

The following is a function with $\alpha$ being a real constant $$f(t) = \frac{\sin(\alpha t)}{\alpha t}.$$ Determine the analytic signal $f_a (t),$ Hilbert transform $\hat{f}(t),$ and the envelope ...
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0answers
28 views

explanation of thm proving an operator is compact [graduate functional analysis]

I've linked to a Theorem (from H&N's Applied Functional Analysis) whose proof I'm trying to understand and I was wondering if anyone could help me out. The theorem is describing how to show that ...
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1answer
16 views

Need a little clarifying on dividing orders of elements of groups

I am reading a little proof here and can't seem to see what theorem they use: Claim: Let G be a group with $a \in G$ and $\text{ord}(a)$ finite. If $H$ is a normal subgroup of $G$, then ...
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2answers
39 views

Is the set of periodic functions from $\mathbb{R}$ to $\mathbb{R}$ a subspace of $\mathbb{R}^\mathbb{R}$?

A function $f: \mathbb{R} \to \mathbb{R}$ is called periodic if there exists a positive number $p$ such that $f(x) = f(x + p)$ for all $x \in \mathbb{R}$. Is the set of periodic functions from ...
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3answers
30 views

Minor flaw in understanding of the proof of the derivative of exponential functions

I understand the majority of the proof of the derivative formula for exponential functions of the form: (full proof at bottom of post) $\frac{d}{dx}a^x$ but I have a little trouble with the last ...
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3answers
27 views

Proving a basis in linear algebra

So at the moment I'm trying to go through proofs and I came across this one: Suppose $ P_n $ is the vector space of all polynomials with degree less than or equal to n. Prove that $ \{1, x − 1, ...
2
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1answer
34 views

Show that for propositional logic $\vdash_i \neg \varphi \Leftrightarrow \vdash_c \neg \varphi$.

As the title says, where $\vdash_i$ is derivations in Intuitionistic logic and $\vdash_c$ is derivations in Classical logic. I am allowed to use a corollary that states that $\vdash_i \varphi ...
2
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4answers
65 views

Prove that if $A$, $B$, and $C$ are sets then $(A - B) \cup (A - C) = A - (B \cap C)$ [duplicate]

Prove that if $A$, $B$, and $C$ are sets then $(A - B) \cup (A - C) = A - (B \cap C)$. I have the proof for the first direction: Let $x \in (A - B) \cup (A - C)$ be given. Hence, $x \in (A - B)$ ...
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0answers
27 views

Need to prove continuity for Intermediate Value Theorem

I'm in real analysis class and I have a question that reads as follows: prove that x^10000 + x^1000 - 1 = 0 has a solution with 0 < x < 1. I'm thinking I ...
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2answers
74 views

Product of negative numbers [duplicate]

Why is a negative number multiplied by a negative number a positive number? I'm trying to know what does multiplying by a negative number mean. If you think of multiplication as a "groups of" ($3 ...
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1answer
20 views

Question on Fourier-Stieltjes transform (in Rudin, p. $15$)

I have a question on an inequality written on the bottom of this page. Let $G$ be a locally compact group and let $\gamma : G \to S^1 \subset \Bbb C$ be a character of $G$. If $\mu$ is a complex ...
3
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3answers
62 views

Proofs in linear algebra

I'm pretty awful at proving linear algebra proofs, I just don't understand how you know what to do or where the information comes from. I have some sample questions below of what I mean, I have no ...
2
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2answers
83 views

Proof that there are infinitely many primes (Euclid)

I was wondering if I could get some insight on my proof. I am in the midst of relearning some number theory and just "writing proofs" in general, and I would like some assistance to see if I am on the ...
0
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0answers
30 views

Prove that $(L[a,b], ||\cdot ||)$ is a normed linear space

I want to prove that $(L[a,b], ||\cdot ||)$ is a normed linear space with norm $$||f(x)|| = \int_a^bx^2|f(x)|dx.$$ First, let $\lambda \in \mathbb{R},$ then it is clear that by properties of ...
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1answer
15 views

$L^{\infty}$ is a normed linear space.

I want to prove that $(L^{\infty}(E), ||f||_{\infty})$ is a normed linear space with norm $$||f||_{\infty}= \inf\{M\geq0 : |f(x)| \leq M \text{ for almost all } x\in E\},$$ where is a measurable set. ...
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0answers
24 views

Linear Transformation representation matrix.

I have some linear algebra question regarding let $\left(\vec{v_{1}},\vec{v_{2},}\vec{v_{3}}\right)$ be a base for$ \mathbb{R}^{3}$ and$ \left(\vec{w_{1}}\vec{w_{2}}\right)$ be a base for$ ...
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2answers
42 views

Does a proof exist for the reflexive property (x=x)?

I have read an article suggesting that proofs or explanations do not exist for some very basic properties in math, including "$x$ is equal to $x$." A preliminary online search did not yield a ...
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1answer
43 views

Proof that there are infinitely many sin values

Prove that there are infinitely many $n\in\mathbb{N}$ such that $sin(n)>\frac{1}{2}$ and infinitely many $n\in\mathbb{N}$ such that $sin(n)<\frac{-1}{2}$. Seems so simple and probably is but ...
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1answer
21 views

Understanding a proof from Rotman's “Advanced Modern Algebra”(Chinese Remainder Theorem)

Please, read this post. I don't need to find any proof of the theorem, a I need to understand a specific step in a stecific proof. This is the proof from J.Rotman's book "Advanced Modern Algebra" 3rd ...
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4answers
51 views

Understanding a proof by induction

In the following proof by induction: Problem: Prove by induction that $1+3+ \ldots+ \ (2n-1)=n^2$ Answer: a) $P(1)$ is true since $1^2=1$ b)Adding $2n+1$ to both sides we obtain: $$ ...
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1answer
15 views

Proving part of a language is not regular [on hold]

I am looking to prove that a language is not regular. In order to use Pumping Lemma, I can only prove that part of the language is not regular. This language in particular will always consist of a ...
2
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1answer
66 views
+50

Proof of algebraic set involving dimension

I need some help to understand the following proof. Let $k$ a field and $V$ an algebraic set. I note $\mathfrak{m}_P$ the ideal generated by $X_1-a_1,\dots ,X_n-a_n$ in $k[V]=k[X_1,\dots ,X_n]/I(V)$ ...
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1answer
51 views

Riemman Sum proof: x^2 [closed]

How can I prove that $$\int _{a}^{b} x^2 = \frac{b^3-a^3}{3}$$ Using the definition of Riemman Sum
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0answers
13 views

Estimating elasticity of y with respect to x in a log-log specification

The question My rudimentary workings so far is that; log(y_i/x_i) = log(y_i)-log(x_i) Factorise, so, log(y_i/x_i) = log(y_i) + upsilon_i - log(gamma_i + 1) Thus, elasticity of y to x is always >1 ...
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2answers
38 views

Question about Leonard Gillman's proof of the divergence of the Harmonic series.

Leonard Gillman (1917 – 2009) was an American mathematician, emeritus professor at the University of Texas, Austin. His proof of the divergence of the Harmonic series appeared in The College ...
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0answers
21 views

Necessity of $C^{1}$ hypothesis in fundamental theorem for line integrals

The statement for the fundamental theorem for line integrals I have in my (unpublished) textbook is: Let U ⊆ Rn be an open set, let φ : [a,b] → U be a piecewise smooth curve, and let $Ω = C_{φ}$. Let ...
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1answer
23 views

Proof explanation - weak law of large numbers

Let $(X_i)$ be i.i.d. random variables with mean $\mu$ and finite variance. Then $$\dfrac{X_1 + \dots + X_n}{n} \to \mu \text{ weakly }$$ I have the proof here: What I don't understand is, why ...
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1answer
37 views

Proof explanation of $[0,1]$ is compact

Let $X=[0,1]$. Prove $X$ is compact. Let $\{U_i\}_{i\in I}$ be an open cover of $X$, or equivalently $$X=\bigcup_{i\in I} U_i~\text{and each}~ U_i~\text{is a open subset of}~X.$$ By definition of ...
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0answers
52 views

I did not understand one thing in the proof of substitution lemma?

The substitution lemma in lambda-calculus is proved by the following way, but I just did not understand the application of induction hypothesis in it. The lemma as shown below, where x and y are ...
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1answer
22 views

$G_{\mathfrak a}(A)$ integral domain and $\bigcap \mathfrak a^n = 0$ implies $A$ is integral domain

This is Lemma 11.23 in Atiyah: For an ideal $\mathfrak a \subseteq A$, define $G_{\mathfrak a} (A) = \bigoplus _{n=0} ^\infty \mathfrak a^n / \mathfrak a^{n+1}$. The statement of the Lemma: ...
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1answer
65 views

Formal Proof that $f^{-1} \circ f = id_x \ , \forall f$

Given $f$ as an invertible function with domain $X$ and codomain $Y$, then we can say $$f^{-1}(f(x)) = x $$ Or since they are both logically equivalent $$ f(f^{-1}(x)) = x $$ This can also be ...
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1answer
28 views

Strong Induction Proof of amounts of money

I am so confused about this kind of question which is referring to amounts of money. I know we should use strong induction to prove if we meet some questions asking you which amounts of money can be ...
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1answer
20 views

Question about the proof that a normed space is finite-dimensional if its closed unit ball is compact

I have a question about a minor point in the proof of the following theorem. Let $B$ denote the unit closed ball in $(V,\|\cdot\|)$. If $B$ is compact then $V$ is finite-dimensional. Proof. ...
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1answer
16 views

Question about proof for bipartite containing no odd cycles

I red the proof from here https://proofwiki.org/wiki/Graph_is_Bipartite_iff_No_Odd_Cycles Sufficient Condition Let $G=(V,E)$ be bipartite. So, let $V=A∪B$ such that $A∩B=∅$ and that all ...
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1answer
32 views

Let $f$ be a differentiable function such that $f(0)=0$. The function $f$ is odd if and only if its derivative, $f'$, is even. [closed]

I need to show both directions of this biconditional statement. Let $f$ be a differentiable function such that $f(0)=0$. The function $f$ is odd if and only if its derivative, $f'$, is even.
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2answers
27 views

Let A be an infinite set, and S a sequence that contains A. Is A countable?

Let $ A $ be an infinite set, and $ S $ a sequence that every element of $ A $ appears at least one time. How can I prove that $ A $ is countable (or uncountable)?
2
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0answers
26 views

help with a proof on Doob's Submartingale inequality - application of chebychev's inequality

I am stuck on a final step of the proof, we have that $(X_n)$ are non negative submartingale, and $c>0$. We let $T = \inf \{n: X_n > c \} \wedge N$ which is a stopping time. Let $E \{ ...
2
votes
1answer
30 views

Prove that the center of a circle within a constructed triangles lies on the angle bisector

I was given steps to construct a figure: 1.) Construct a horizontal ray AB and a segment AC at an angle to the ray. Locate point D anywhere on ray AB and construct the segment CD. 2.) Construct the ...
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2answers
26 views

Unclear proof of irreducibility on $x^m+1$

I came across this problem and frankly, it's unclear looking at the solution. Prove $x^m+1$ irreducible in $\mathbb{Q}(x)$ if and only if $m=2^k$ for $k \in \mathbb{N}$ Well the solution's short ...
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0answers
28 views

Proof of divisibilty in the natural numbers.

The problem: Suppose that c ≥ 2 is a natural number which is not prime. Show that there is a natural number d ≥ 2 such that d|c and d ≤ √c. Been stuck on this for ages. Not really sure how to get ...
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0answers
31 views

Proving the Pigeonhole Principle

I am looking to prove the Pigeonhole Principle by proving the following claim: Let $A$ be a set with $m$ elements, and let $B$ be a set with $n$ elements, where $m,n\in \omega$ and $m > n$. ...
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0answers
22 views

why absolute values are used for convergence of a limit

This is the definition i am using for convergence of a limit: For all $\epsilon$ > 0 there exist N $\in$ (the natural numbers) $ N $ such that |$a_n -l | < \epsilon $ for all n > $ N $. My ...
3
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4answers
116 views

Mathematical Rigor in Proving Limits by $\epsilon-\delta$ Definition

I am trying to find the most mathematically rigorous way to prove limits, using the $\epsilon-\delta$ definition of a limit, so far I have found two clear cut methods of proving limits using the ...
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2answers
40 views

Actuarial Mathematics proof error?

Is there something wrong with this proof or is it just me? Did they forget the -ve sign or does it cancel somehow? (From the second to last to last line)
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3answers
45 views

If $f:B \to A$ is a surjection, then there exists an injection $h : A \to B$ such that $f \circ h = I_A$

Let $B$ be a finite, nonempty set and assume that $f : B \to A$ is a surjection. Prove that there exists a function $h : A \to B$ such that $f \circ h = I_A$ and $h$ is an injection. ($I_A$ is the ...
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1answer
52 views

Proof that $\liminf_{x\to\infty}f(x) \leq \limsup_{x\to\infty}f(x)$.

I can't understand this proof from my old lecture notes. $\liminf$ is defined as: \begin{align*} \liminf_{z\to\infty}f(z) = \inf_{x< y} \sup_{y< z}f(z) \end{align*} and $\limsup$ is defined ...
2
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1answer
27 views

Lemma characterizing second fundamental form, do not understand step

Consider an excerpt of a lemma and part of its proof from a Riemannian geometry textbook. Lemma. The second fundamental form is independent of the extensions of $X$ and $Y$; bilinear ...
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2answers
39 views

Need help understanding part of the proof that $\displaystyle\int_{x=-1}^{1}[P_L(x)]^2\,\mathrm{d}x=\frac{2}{2L+1}$

I am struggling to understand some of the proof that $\displaystyle\int_{x=-1}^{1}[P_L(x)]^2\,\mathrm{d}x=\frac{2}{2L+1}\tag{1}$ In my book I have a list of $6$ recursion relations for Legendre ...
3
votes
1answer
66 views

Proving $\lim_{x\to 1}\frac{2+4x}{3}=2$ using the $\epsilon$ -$\delta$ definition of a limit

I'm looking for a verification of my $\epsilon-\delta$ proof of a limit example, if my proof is not completely mathematically rigorous, please tear it apart. Required to Prove: $$\lim_{x \to 1} \ ...