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

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13 views

Existence of nice exhaustion - Rudin.

This is taken from Rudin's Complex Analysis/Real Analysis Can someone tell me why $K_n \subset \Omega$? I agree it is compact, but why does it follow that it is a subset of $\Omega$? WLOG, I ...
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42 views

Complex/Real Analysis mysterious quantity.

The following is a lemma to prove the Runge (only excerpt) page 629 in the link Can someone explain how they got the $$\frac{b_1 - b}{(z-b)}.$$ I believe he took $$\frac{1}{z - b}$$ and the other ...
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1answer
58 views

On the associative property of a binary operation of the fundamental group.

I was reading about the proof of associativity property of the operation on the fundamental group here. The book gives the following diagram then it says the reader should supply the elementary ...
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30 views

Generic freeness (a Lemma from Matsumura, CRT)

Let $B$ be a Noetherian ring, and $C$ a $B$-algebra generated over $B$ by a single element $x$; let $E$ be a finite $C$-module, and $F\subset E$ a finite $B$-module and $CF=E$. Then $D=E/F$ has an ...
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49 views

Give me the proof of this equality!? [duplicate]

I would like that someone can tell me stap for stap (mathematically proof) that this equality is true! I hope someone can help me and if you can I would be very thankfull ;) ...
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0answers
39 views

Minor issue about the proof of the Cauchy Convergence Criterion on Understanding Analysis (Abbott)

I don't understand one small part of the proof of the following statement: If a sequence is a Cauchy sequence, then it converges. PROOF: Let $(x_n)$ be a a Cauchy sequence. Then it is bounded ...
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1answer
21 views

Discrete Math - Relation among the relations

I need help understanding an assignment involving sets. We're given a few sets, $A=\{1,2\}$, $B=\{1,2,3\}$, $C=\{1,2,3,4\}$, $P=\{(1,1),(2,3),(3,2)\}$, $Q=\{(1,1),(2,2),(3,2)\}$, ...
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0answers
23 views

blocking set of projective planes

Have a quick question, considering the Desarguesian projective plane of order q, what is an upper bound of the minimal blocking set? Wikipedia says the size of the blocking set is bounded below by ...
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1answer
38 views

Explanation of a proof about continuity in Spivak's Calculus

I can't see the "it follows that" part of the following proof in Spivak's Calculus book: Given that $f$ is continous at $b$, there is a $\delta>0$ such that; if $|x-b|<\delta$ then ...
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1answer
40 views

Regarding the proof of $\int_0^xf(u)(x-u)du$

This question has already been asked: For continuous function $f$, prove: $\int_{0}^{x} \; \left[\int_{0}^{t}f(u) \;du \right] \;dt=\int_{0}^{x} f(u)(x-u)du$ but I really don´t understand the part ...
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2answers
49 views

$\det(aI-T) = 0$ implies $a$ is an eigenvalue of $T$

In Hoffman and Kunze's linear algebra, right after the definition of an eigenvalue for an endomorphism $T: V \to V$ (i.e. $a\in F$ is an eigenvalue if there exists $\alpha\in V$, $\alpha \neq 0$, such ...
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1answer
25 views

Not understanding a line in a proof concerning Monomorphism and injectivity

In the proof that "in the category Div of divisible (abelian) groups and group homomorphisms between them there are monomorphisms that are not injective" given in Wikipedia ...
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1answer
74 views

No cycle containing edges $e$ and $g$ implies there is a vertex $u$ so that every path sharing one end with $e$ and another with $g$ contains $u$

There is a proof in my textbook for the following claim, which doesn't make a whole lot of sense to me. My annotations are in bold. Could someone perhaps elaborate on what's going on? Claim. If ...
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0answers
34 views

Theorem 2 of Section 5.4 in Fulton's Algebraic Curves

The Theorem and its proof are: To understand the proof we also need: Finally, Corollary 1 for Bezout's Theorem says that $\sum_{P \in F \cap G} m_P(F) m_P(G) \le deg(F) deg(G)$ for any curves ...
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1answer
60 views

Prove that the Zariski space $\text{Zar} \space (K,A)$ is compact.

I posted part of the proof from Matsumura's Commutative Ring Theory. I got stuck in the last sentence where it says "Hence the intersection of all the elements of $\mathcal{A}$ is the same thing as ...
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0answers
66 views

Bass' paper on Gorenstein rings

I am currently reading the paper On the ubiquity of Gorenstein rings by Hyman Bass. I found difficulty to understand the proof of Proposition (7.2). Under the the following setting: $A$: commutative ...
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2answers
28 views

Let $F$ be a class of sets. Prove that $B - \mathop{\bigcup}_{A\in F} A = \mathop{\bigcap}_{A\in F} (B-A)$

Let $F$ be a class of sets. Prove that $B - \mathop{\bigcup}_{A\in F} A = \mathop{\bigcap}_{A\in F} (B-A)$ I've started like this: $X= B - \mathop{\bigcup}_{A\in F} A$ $Y= \mathop{\bigcap}_{A\in ...
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4answers
72 views

Prove that if $a<b$, then $-b<-a$

Prove that if $a<b$, then $-b<-a$ I'm a bit lost in this one, this is what I did: First case: $0<a<b$ $|a|<|b|$, so $|-a|<|-b|$ Since both are negative and $|-b|$ is greater ...
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1answer
112 views

How to remember all the proofs in mathematics

I have a problem where I forget the proof of a theorem after some time without reworking it out. However, my teacher said that he was able to prove a theorem even without reworking it out for a long ...
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1answer
43 views

Ten men selected to include a captain vs ten men selected to include at least one officer

I can't see how these are different scenarios, but the answers are different. The general scenario is that there is a company of volunteers that consists of a captain, a lieutenant, an ensign, and ...
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0answers
30 views

Help on Lagrange Error Calculation [duplicate]

Here is an example in Burden's Numerical Analysis book. My problem is in bold In example 2 we found the second Lagrange polynomial for $f(x)=1/x$ on $[2,4]$ using the nodes $x_0=2$, $x_1=2.75$, ...
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2answers
112 views

Confusion on the proof that there are “arbitrarily large gaps between successive primes”

I am trying to wrap my brain around a proof that proves that there are arbitrarily large gaps between successive primes. The proof is Given a natural number $N\ge2$, consider the sequence of $N$ ...
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2answers
37 views

How can the only maximal ideal of $C[x] / X^2$ be $(X)$?

In my notes I have the following example which I don't understand. Let $f$ be the canonical injection from $C$ to $C[X]/X^2$.The only maximal ideal of $C[X]/X^2$ is $(X)$ and $f^{-1}((X))$=$(0)$. ...
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1answer
73 views

Help needed to understand statements about torus

I am having trouble understanding two statements: Let $A$ be an algebraic curve in $\mathbb{P}^2$ over $\mathbb{C}.$ Consider its normalization $$\pi: \hat{A} \to A.$$ If genus $g(\hat{A})=1,$ ...
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2answers
84 views

Proof of the Intermediate Value Theorem [closed]

Theorem: Let $f$ be continuous on $[a,\,b]$ and assume $f(a)<f(b)$. Then for every $k$ such that $f(a)<k<f(b)$, there exists a $c\in[a,\,b]$ such that $f(c)=k$. proof: $f$ continuous at ...
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0answers
47 views

The proof of remainder theorem for polynomials: $f(\lambda)=0$ if and only if $x-\lambda$ divides $f$

From the paragraph starting with "Let us now suppose that $f(\lambda) = 0$" the author states that if $\deg(f) \leq 0 $ then $f=0$. But if $\deg(f) = 0$ then $f$ can be any constant, and not ...
0
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1answer
153 views

strong induction example

There is following example given in a book. I am not sure how do we conclude that $a$ is divisible by prime? See this section: Case 2 ($k + 1$ is not prime): In this case $k+1=ab$ where $a$ and ...
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2answers
50 views

Explanation of this proof of the principle of analytic continuation

I am confused about the last paragraph, He states that we choose a point $z_1$ inside the disk, and draw another disk around $z_1$ with radius $\delta_1$ Then says "f is identically zero on this ...
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1answer
29 views

Principle of analytical continuation proof confusion

Let $f$ be holomorphic on a region $D$ and assume I can find a sequence $\{w_k\}$ in $D$ s.t. $w_k \to z_0$ w/ $z_0 \in D$, $w_k \not = z_0$ & $f(w_k) = 0$. Then $f(z) = 0$ $\forall z \in D$ ...
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1answer
44 views

Fundamental Characterisation of Measureable Functions Explanation (Characterization)

I am just starting learning about measure theory (what a great way to spend Christmas...!), and I am unclear on this following claim: Characterisation of Measurable Functions. It appears to be a very ...
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3answers
82 views

Just a proof of algebra

If $a+b+c=0$, Show that $\left[\dfrac{a}{b-c}+\dfrac{b}{c-a}+\dfrac{c}{a-b}\right]\left[\dfrac{b-c}{a}+\dfrac{c-a}{b}+\dfrac{a-b}{c}\right]=9.$ I am struck with this problem but can't find a ...
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2answers
62 views

In Zagier's one-sentence proof, why is S defined to be {(x,y,z)∈ℕ^3:x^2+4yz=p,p prime}?

I've looked at a very clear explanation of Zagier's proof (specifically, it can be found here:http://danielmath.wordpress.com/2012/12/26/one-sentence-proof/) but the first step still eludes me: why is ...
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1answer
57 views

Localization of Rings: Show $R_{f} \simeq R[X]/\langle 1-fX \rangle$

Let $R$ be a ring, $f \in R$, and $X$ a variable. Show $R_{f} \simeq R[X]/\langle 1-fX \rangle$. I am a beginner in algebra and I am reading a textbook in commutative algebra. What I do not ...
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1answer
52 views

Explanation of Proof that field sum of more than 2 elements is 0. [duplicate]

"Suppose the field $F$ is finite. If $f\colon F\to F$ is any bijection, then we can conclude that $\sum_{x\in F}x=\sum_{x\in F}f(x)$. Let $\alpha\in F$ such that $\alpha\ne 0$. Then $x\mapsto \alpha ...
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0answers
62 views

Is proper morphism from affine scheme affine?

I'm reading Mumford-Oda's lecture notes http://www.math.upenn.edu/~chai/624_08/mumford-oda_chap1-6.pdf. And they use the fact:"Let $f:U \to Y$ be a proper morphism of noetherian schemes and $U$ is ...
2
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2answers
80 views

Prove that the unity element in a subfield of a field must be the unity of the whole field

The solution I was given says: Let $F$ be a field and suppose $u^2=u$ for some nonzero $u$ in $F$. By multiplying each side by $u^{-1}$ it is clear that $0$ and $1$ are the only solutions of ...
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1answer
40 views

Explaining That there is no nontrivial ring homomorphism between Z and nZ

My instructor wrote in his notes the following example: "As groups (Z,+) and (nZ,+) are isomorphic. As rings is there any nontrivial homomorphism $\phi$: Z->nZ? The answer is no and he gives the ...
5
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1answer
98 views

Construction of a triple cover of $A_6$ in “Finite Simple Groups” by Wilson

I am reading The Finite Simple Groups by Robert Wilson: see page 29. I want to understand a construction of triple cover of $A_6$. On section 2.7.3., I don't understand the second paragraph, which is ...
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1answer
73 views

Satellite functors in Cartan Eilenberg

I was reading and came across this statement whose proof is said to be obvious. I however after hours still cannot figure out how to prove $S_2T(A) = S_1(S_1T(A)) = S_1T(M)$. The definitions are: ...
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1answer
63 views

Doubt on a paragraph regarding Lagrange's multiplier.

I've a topic in my notes "The method of Lagrange's multipliers" which is described as follows: Let $U$ be an open set in $\mathbb R^n$.Let $f\in C^1(U,\mathbb R)$ and let ...
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1answer
38 views

Question on abstract algebra about Group?

I need an explanation, why $ (\mathbb{Z}_7,\oplus _6 )$ is not a Group? As I have discovered so far. The following conditions are satisfied I) Closed! II) Associative! III) ...
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1answer
19 views

Sequence Lemma explanation

Then every neighbourhood $U$ of $x$ contains a point of $A$. So I don't see it happening unless $X$ is a metric space, but the proof is for any topological space.
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1answer
31 views

Haudorff Formula Set Theory

For every $\alpha$ and every $\beta$, $$\aleph_{\alpha+1}^{\aleph_{\beta}}=\aleph_{\alpha}^{\aleph_{\beta}} \cdot \aleph_{\alpha+1}$$ Proof: If $\beta \geq \alpha+1$, then ...
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1answer
25 views

Question regarding an algebraic manipulation in GFology

How does the author arrive at the last equality in the first line, i.e.$$\text{why is } [x^k]\frac{1}{1-y(1+x)} = \frac{1}{1-y}[x^k]\frac{1}{1-\left(\frac{y}{1-y}\right)x} \text{?}$$
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2answers
62 views

set theory proof explanation

The following lemma is taken from the book 'Introduction to Set Theory' by Hrbacek and Jech. chapter $6$ normal form Can anyone explain to me why the first sentence holds ( the existence of ...
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0answers
29 views

Multilinear Function proof in Spivak?

Note that $$\wedge^n (V)$$ denotes the set of all alternating multilinear functions and $\mathfrak{I}^n(R^n)$ denotes the set of all multilinear function. I don't know what the actual symbol ...
2
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1answer
32 views

Multilinear algebra and matrices

Given $\wedge^k(V)$ an alternating multilinear space and $T : V \to W$ a linear map, then we have $$v_1 \wedge \dots \wedge v_k \in\wedge^k(V).$$ Define $$\wedge^k(T)(v_1\wedge\dots\wedge v_k) = ...
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1answer
60 views

Understanding the proof that $c_0$ is a closed subspace of $\ell^\infty$

The problem is given: source Let $c_0$ be a space of real sequences $x = \{x_n\}_{n = 1}^\infty=0$ converging to $0$. Let $\ell^\infty$ be a set of real sequences $w = \{w_k\}^\infty_{k=0}$ ...
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1answer
82 views

A limit theorem in Rudin. Please elaborate?

Theorem: Let $\{ p_n \} \in X $. If $E \subset X$ and $p$ is a limit point of $E$, then there is a sequence $\{ p_n \}$ such that $p = \lim p_n $ The proof goes like this proof: For each ...
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0answers
41 views

Transfinite Induction (Proof Explanation)

The theorem above is extracted from the book 'Introduction to Set Theory' by Hrbacek and Jech. Questions: $1$)I don't understand the successor case. When $\alpha_2=\beta+1$, why suddenly $W_2$ ...