Tagged Questions

For questions about approaches and techniques for discovering a proof, as opposed to writing it down clearly (which involves (proof-writing)). Should not be used unless the focus is on the technique of the proof instead of the solution.

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Prove that the last process is responsible for $\mathrm\lceil {n\over p} \rceil$ elements

In my parallel computing class, I have this proof in one of my HW problems: Suppose we have chosen a block agglomeration of $\mathrm n$ elements (labeled $\mathrm 0, 1, ... , n - 1$) to $\mathrm p$ ...
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Suppose that the function f:(a,b) --> R is uniformly continuous. Prove that f:(a,b)-->R is bounded.

Also a second unrelated question, suppose that the function $f:[0,1] \to \Bbb{R}$ is continuous with $f (0)>0$ and $f (1)=0$. Prove that there is a number $x_0 (0,1]$ such that $f (x_0)=0$ and ...
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Working on an inequality

Hey all I am trying to show that an inequality involving integrals is true, and I am not sure if I am taking the right approach or what I am missing. I will write one of the inequalities I am trying ...
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Prove compact sublevel sets imply coercivity

$f$ is convex and $dom(f):x \in \mathbb{R}^N$. Define sublevel sets of $f$ as $$\mathbf{S}(f,\beta)=\{x \in \mathbb{R}^N\ : f(x) \leq \beta \}$$ are compact. I need to ...
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How to prove this form of $n$?

Show that every positive integer is a sum of one or more numbers of the form $2^r3^s,$ where $r$ and $s$ are nonnegative integers and no summand divides another. From: AOPS Putnam A1 Solution I ...
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Maximal Principle: Why using the new transition matrix $\tilde{P}$?

First some notation: Let $(X,E,P)$ denote a finite, irreducible Markov chain with finite state space $E$ and transition matrix $P$. Choose and fix a subset $E^°$ of $E$, which will be called ...
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prove there is no smallest positive rational number

How would I prove there is no smallest positive rational number? what is the best method to prove this statement?
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Prove $1+2\sqrt3$ is not a rational number

How would I go about proving $1+2\sqrt 3$ is not a rational number assuming $\sqrt 3$ is not a rational? Would direct proof be the easiest? Total beginner here, any insight would be appreciated.
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Best way to prove adjacency of a graph

Assume n is even. Considering a graph where each vertex in $v_1,...,v_n$ is adjacent to the next (ie $v_i \sim v_{i+1}$ for $1\leq i<n$) and where $v_1,v_n$ are each connected to at least $n/2$ ...
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Prove if $n^2$ is even, then $n^2$ is divisible by 4

I am working on this question Prove for every integer n if $n^2$ is even, then $n^2$ is divisible by 4. prove by contradiction Proof: Since there exists an integer $n$ such that $n^2$ is ...
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Proof: Finite product of elementary matrices.

I need to proof the following theorem: Let $A,B \in M_{mxn}(\mathbb{R})$ and $B$ is $A \sim B$ (equivalent of rows with $A$), then there exist a matrix $C \in M_{m}(\mathbb{R})$, such that $B = CA$, ...
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Let $f$ have a jump discontinuity at $x_0$. Show that $f(x_1), f(x_2), \ldots$ has at most two limit-points.

This is a question I understand intuitively but am having trouble proving rigorously: Let $f$ have a jump discontinuity at $x_0$. Show that if $x_1, x_2, . . .$ is any sequence of points in the ...
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Simple Division Proof

Prove that for every three integers i, j, and k, if i $\nmid$ jk, then i $\nmid$ j We've just started proofs and I am at a complete loss for how to go about doing it. I've tried proving through ...
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Surjective linear transformation in $\mathbb R^2$ is injective

Let $T:\mathbb R^2 \to \mathbb R^2$ be a surjective linear transformation. Prove that $T$ is injective without using the rank-nullity theorem Is there a way to prove this theoremn, without the ...
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Lemmas to characterize Eistenstein primes

Here are lemmas to characterize Gaussian primes Lemma1 Let $p$ be an odd prime such that $p\equiv 1 \pmod 4$. Then, the Legendre symbol $(-1/p)=1$ Lemma2 Let $p$ be an odd prime such that ...
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Proving an Inequality

Suppose I've got a positive number $a$>0. I've got some other positive number $b$>0. I am given that $a$ > $b$. How do I prove that the above implies $-a + \sqrt{ a^{2} - b^{2} }$ < 0 ? So far I ...
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squeeze theorem - math

I am trying to prove the following: 1/ek <= (1/k)(1-(1/k))^(k-1) <= 1/2k for k>=2 in doing so I tried induction proof, and contradiction and it didn't work, it gets too complicated... Then ...
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Establish the convergence or divergence of a sequence [duplicate]

Establish the convergence or divergence of the sequence (y_n), where: y_n := 1/(n+1) + 1/(n+2) + ... = 1/(2n) for n /in N.
Proving $a^{(p-1)p^{k-1}} \equiv 1 \pmod {p^k}$ without Euler's Theorem
Is there a different way of solving $$a^{(p-1)p^{k-1}} \equiv 1 \pmod {p^k}$$ without the use of Euler's Theorem (or proof of Euler's theorem for $p^k$)? I've tried to use the Chinese Remainder ...
Let $U\subset \mathbb{R}^{n}$ and $f:U\to \mathbb{R}^{n}$ injective and class $C^{1}$ such that $\det f'(x)\not=0$ for all $x \in U$. Show that $f(U)$ is open and $f^{-1}:f(U)\to U$ is ...