# Tagged Questions

For questions about the formulation of a proof, not about the mathematics behind it.

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### Prove that the min and max of 2 continuous function are continuous

Prove that if $f$ and $g$ are continuous functions the so are min⁡{f(x),g(x)} and max⁡{f(x),g(x)} I know this is true when $f$ and $g$ are not intersect each other, then I can compare them. However, ...
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### Proof related to Harmonic Progression

The question is as follows: Let $m_1<m_2<m_3<\cdots<m_k$ be postive integers such that $\frac{1}{m_1}$, $\frac{1}{m_2}$, $\frac{1}{m_3}$, $\cdots$, $\frac{1}{m_k}$ are in arithmetic ...
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### How to prove in $r_1p_1 +r_2p_2 =u\gcd(p_1,p_2)$, $u$ is a uniformly random polynomial.

Hypothesis: All polynomials are defined over a finite field $\mathbb{F}_p$, where $p$ is a large prime number (e.g. 128-bit prime number). Assume we have two fixed polynomials $p_1$ and $p_2$ of ...
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### Derivative of Exponential map on manifolds

I'm trying to compute the derivative of the map $f:\Sigma\times [0,\delta)\to M$ given by $$f(p,t)=\exp_p tN(p),$$ in $X\in T_p\Sigma$, where $(M^n,g)$ is a Riemannian manifold, $\Sigma\subset M$ a ...
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### How to describe “the digits of n”

Prove that: 6 divides n if and only if 6 divides the sum of the digits of n Solution n mod 3 = 0 iff (sum of the digits of n) mod 6 = 0 How would one represent (the digits of n)?
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### Proof idea: Let $(X,d)$ be a metric space, and $\rho$ be bounded metric, show that they will generate the same topology

Let $(X,d)$ be a metric space, $d$ generates the metric topology $\mathcal{T}$ via metric ball $B_\epsilon(x)$. Show that bounded metrics: $\rho_1(x,y) = \dfrac{d(x,y)}{1+d(x,y)}$ with ...
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### Inductive Proof for Vandermonde's Identity?

I am reading up on Vandermonde's Identity, and so far I have found proofs for the identity using combinatorics, sets, and other methods. However, I am trying to find a proof that utilizes mathematical ...
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### What is a proof?

I am just a high school student, and I haven't seen much in mathematics (calculus and abstract algebra). Mathematics is a system of axioms which you choose yourself for a set of undefined entities, ...
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### Trouble Proving that if $f : A \rightarrow B \text{ then } I_{B} \circ f=f$

Proving that if $f : A \rightarrow B \text{ then } I_{B} \circ f=f$ My problem with this question is that I do not know how one derives the theory in order to get the correct answer. I will ...
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### extension of Cauchy's Integral formula

This question is from Brown and Churchill's Complex Variables and Applications, 8ed., Section 52, Question 6. Let $f(s)$ denote a continuous function taken along a simple contour, $C$ enclosing a ...
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### Logarithm proof problem: $a^{\log_b c} = c^{\log_b a}$

I have been hit with a homework problem that I just have no idea how to approach. Any help from you all is very much appreciated. Here is the problem Prove the equation: $a^{\log_b c} = c^{\log_b a}$ ...
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### A simple proof in the form of an inequality

Proof that for all $a, b$ are elements of $\mathbb{R}$ : $(a+b)^2\geq 4ab$. Does it satisfies after doing some simple arithmetic to say that $(a-b)^2\geq 0$? Or do I need to go over all the cases ...
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I'm trying to prove that the limit of the sequence $x_n^k$ is the same as the limit of $x_n$ all raised to the $k$th power. Prove $$\lim _{ n\rightarrow \infty }{ { x }_{ n }^{ k } } ={ \left( \lim ... 1answer 39 views ### Fisher information for exponential family: Regularity conditions for the Fisher-Information to be defined certain regularity conditions have to be fulfilled (like in Lemma 5.3. in Theory of Point Estimation by E.L. Lehmann or on slide 2 here: http://www.stat.nus.... 0answers 29 views ### How to calculate \Delta in conic sections? When learning conic section I learnt about \Delta. For any conic in general form : ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 Here \Delta=abc +2fgh - af^2 - bg^2 -ch^2 The conic is said to be ... 2answers 43 views ### If \operatorname{Mod}(T_1 \cup T_2) = \emptyset then for some \sigma, T_1 \vDash \sigma and T_2 \vDash \neg \sigma Problem description: if T_1 and T_2 are theories such that \operatorname{Mod}(T_1 \cup T_2) = \emptyset, then there is a \sigma such that T_1 \vDash \sigma and T_2 \vDash \neg \sigma. I ... 1answer 41 views ### Proving the Mean Value Theorem So I'm working through some questions in my book and I don't understand how to finish out the attached problem. For the first 5 blanks here is what I got: Since f is continuous on [16,25] and ... 1answer 39 views ### f is injective if and only if for all y \in B there exists at most one x \in A such that f(x)=y Let A and B be sets and let f:A \to B be a function. Assume f is injective. Let y \in B. There are two cases to consider. If there exists an x \in A such that f(x) = y, then x is ... 14answers 2k views ### How are proofs formatted when the answer is a counterexample? Suppose it is asked: Prove or find a counterexample: the sum of two integers is odd The fact that 1 + 1 = 2 is a counterexample that disproves that statement. What is the proper format in which ... 2answers 68 views ### A **proof** for \sum_{i=0}^{t-2}{\frac{1}{t+3i}} \leq \frac{1}{2} [duplicate] I need a proof for the inequality: \sum_{i=0}^{t-2}{\frac{1}{t+3i}} \leq \frac{1}{2} for all natural numbers t \geq 2. For t=2 both sides are equal. Can someone find a proof for all t? maybe ... 2answers 76 views ### Prove integral equality  \int_{0}^{\pi} xf(\sin(x))dx = \pi \int_{0}^{\frac{\pi}{2}}f(\sin(x))dx  [closed] How can I prove the following claim for any given continues function:$$ \int_{0}^{\pi} xf(\sin(x))dx = \pi \int_{0}^{\frac{\pi}{2}}f(\sin(x))dx $$Thanks! 2answers 34 views ### Hints on showing that a metric space is complete Let C[0,K] be the space of all continuous real valued functions on [0,K] for K>0 and L\geq0, equipped with the metric d defined by$$d(f,g)=\sup_{0\leq k\leq K}e^{-Lk}|f(k)-g(k)|. I ...
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I'm in the process of writing my Master's Thesis on automata theory. The writing must be in English which isn't my mother tongue. So the question is, given that this is my first time long (hundred ...
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### Prove that $4$ divides $3^{2m+1} - 3$

Prove that $4$ divides $3^{2m+1} - 3$. By plugging in numbers I can see this is true, but I can't figure out a way to prove this, I was thinking maybe proving first that it is divisible by $2$, and ...
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### Disproving existence statements

I am trying to get some practice on disproving existence statements and I was really stuck on this one: "There exists an example of three distinct positive integers different from $a,2a,3a$ for some ...
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### Show that the closure of $A$ is the intersection of all closed sets containing $A$, topology proof needed

I want to show that given $(X, \mathcal{T})$, we define $\overline A = \{x \in X| \forall U \in \mathcal{T}, x \in U \implies U \cap A \neq \varnothing\}$ (definition of closure from Munkres), then ...
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### Help in proof: a connected graph is $k$ edge connected iff all blocks are

Attempt: we know that the edge set of $G$ is the union of those of it's blocks (maximal connected subgraphs of $G$ not having a cut vertex), any two of them touching in at most one vertex. If all ...
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### Proving a theorem, what is meant by sufficiency and necessity?

I am looking at the proof of a theorem and the proof begins by saying ...is the proof of the sufficiency part of this theorem so we just need to establish the necessity of the condition. What ...
Using the usual definition of limits, with "epsilon and deltas", how can I show that if $x=(x_1,\dots,x_n)$ is a vector in $R^n$, and $f\colon J\to R$,where $R$ is the set of real numbers and $J$ is a ...
Let $A = \left\{x \in \mathbb{Z} \mid \exists a\in\mathbb{Z}: x = 6a + 4\right\}$ and $B = \left\{y \in \mathbb{Z} \mid \exists b\in\mathbb{Z}: y = 18b - 2\right\}$ and \$C = \left\{z \in\mathbb{Z} \...