# Questions tagged [proof-explanation]

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

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### Proof from Lang’s Basic Mathematics

Starting on page 11, I don’t understand the proof. If a, b are negative integers, then a + b is negative. Proof. We can write a = —n and b = —m, where m, n are positive. Therefore a + b = —n — m = —(n ...
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### Meaning of symbol in probability theorem

What does the $\big|_{t=0}$ mean in this context?
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### Spivak Calculus Ch. 11, Prob. 61a: $f$ diff in interval containing $a$, $f'$ discont. at $a$. Prove one-sided limits of $f'$ at $a$ cannot both exist.

The following is a problem from ch. 11 of Spivak's Calculus Suppose that $f$ is differentiable in some interval containing $a$, but that $f'$ is discontinuous at $a$. Prove the following: (a) The ...
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### $f$ cont. at $a$, $f'$ exists in interval containing $a$ (except possibly at $a$), $l=\lim\limits_{x \to a^+} f'(x)$ exists. Does $f'(a)=l$?

This question regards the following theorem (as stated in Spivak's Calculus): Theorem 7: Suppose $f$ is continuous at $a$, $f'(x)$ exists for all $x$ in some interval containing $a$, except perhaps ...
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### uniqueness of clutching decomposition Hatcher p.$47$ Lemma $208$

In the following accepted questions on mse (first and second) concerning K theory, in particular the uniqueness of the splitting given also in Hatcher p.$47$ Lemma $208$ is addressed by the same ...
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### Mistake in proof of "double derivative test" in calculus textbook

I'm currently studying for a semester test in advanced calculus, and one of the topics covered is finding the local minima and maxima of a 3 dimensional surface. The first theorem that was proved was ...
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### Prove that every cycle graph $C_n$ has $n$ edges

I need to prove this directly and by induction. I do not even know where to start. Question: A cycle graph $C_n$ is a connected graph with $n$ vertices, such that each vertex is adjacent to exactly ...
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### Spivak Calculus: $f$ satisfies $f''(x)+f'(x)g(x)-f(x)=0$, for some g. Prove that if 𝑓 is 0 at two points, then 𝑓 is 0 on the interval between them.

The following is a problem from chapter 11, "Significance of the Derivative" from Spivak's Calculus Suppose that $f$ satisfies $$f''(x)+f'(x)g(x)-f(x)=0\tag{1}$$ for some function $g$. ...
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### Showing that the intrinsic mean of continuously distributed data on the unit circle is almost surely unique

The article I am currently reading is Intrinsic Means on the Circle: Uniqueness, Locus and Asymptotics by Hotz and Huckerman, pp. 7. Below is a screenshot of the authors' proof that the intrinsic mean ...
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### Figuring out a proof that $\mathbb{P}\{-\pi \leq X \leq -\pi + \delta\}=\frac{\delta}{2\pi}+\frac{\delta^{k+1}}{(k+1)!}f^{k}(-\pi+) + o(\delta^{k+1})$

The article I am currently reading is Intrinsic Means on the Circle: Uniqueness, Locus and Asymptotics by Hotz and Huckerman, pp. 4-5. Suppose that $X$ is a random variable living in the unit circle ...
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### Husemoller: homotopy of linear clutching map (proposition $4.5$, pag. $187$)

Background : I'm currently studying vector bundle through the book of [husemoller,"fibre bundles"] (https://www.maths.ed.ac.uk/~v1ranick/papers/husemoller). The following question concerns a ...
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### How to prove $\frac{1}{\sqrt{a} + \sqrt{b}} = \sqrt{a} - \sqrt{b}$?

My daughter is learning how to rationalise surds for her school exams. One example being worked through is the following: $\frac{1}{\sqrt{6} + \sqrt{5}}$ In the tutorials she is following, the first ...
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### Probability of alternating heads/tails - infinite coin tosses

We toss a fair coin until we get two consecutive heads or two consecutive tails. Each sequence of $n$ coin tosses has a probability of $\frac{1}{2^n}$. What is the probability that this experiment ...
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### Proving that an associative binary operation gives rise to a group [duplicate]

I'm trying to prove the following claim. Let $S$ be a nonempty finite set, equipped with an associative operation $*: S \times S \to S$ such that, for every $x,y \in S$, there exists $z \in S$ such ...
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### Understanding a proof about $\lambda$-supercompact cardinal

I have trouble understanding the proof of this Lemma 20.15 from Jech's Set Theory, could someone explain why is $(2^\alpha)^M = (\alpha^+)^M = \alpha^+$?
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### Artin, Chapter 2, Misc.6

I am trying to solve miscellaneous exercise 6 in Chapter 2 of Artin's book, Algebra. Below is the statement of the problem. Let $a = (a_1, \ldots, a_k)$ and $b = (b_1, \ldots, b_k)$ be points in $k$-...
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### Hyperplane arrangement : The Shi arrangement

I have been lately reading Hyperplane arrangement lectures by Richard Stanley on https://www.cis.upenn.edu/~cis610/sp06stanley.pdf . In lecture 5, Theorem 5.16 we define the characteristic polynomial ...
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When using integration by parts, the proof appears to turn the latter function $(x-t)^k$ into the additive inverse of its integral $-(x-t)^{k+1}/(k+1)$. I don't understand how this could possibly be ...