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Prove that min(S) does not exist in S=(0,1)

I'm taking the proof by contradiction route i.e. assuming m = min(S) then trying to find some sort of contradiction. I've tried take m=2m-1 and take m = (m-1)/2 but neither seem to work?
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Show that the ratio between two numbers is always proportional to the the maximum percentage error of their average?

The question is this: if we have a set of any random consecutive numbers, for example {1, 1.2, 4.2, 4.8, 5.6, 7.4, 9.8} then how can we prove that calculating the ratio between each of the numbers and ...
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What is wrong with this proof about the maximizer of a quadratic equation?

Working through Daniel Solow's "How to read and do proofs", I have been stuck at the following problem. Problem: What is wrong with the proof below for the statement. If a, b, and c are real ...
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'Unhappy with my proof' problem: Connectactoe

Connectactoe is tictactoe played in a $3\times3$ Connect4 grid, so gravity plays a part. Player 1 wins if they go in a corner and Player 2 doesn't go in the other corner (see EDIT) when it is a draw: ...
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l’Hôpital’s rule to prove that $f \in \omega(g)$

Can anyone give me a hint on how to start this please? For $n \in R >1$ let $f(n) = n^{4/3}$ and $g(n) = n · (log$5$n)^2$. Use l’Hôpital’s rule to prove that $f \in \omega(g)$.
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If $r$ is a nonzero solution $x^2 + ax + b$, prove that $r | b$

I know that if $r$ is a solution, then there exist two factors of $b$ that when multiplied equal $b$ and that $r$ is one of them. So clearly $r$ divides $b$, but I don't know if there is any other way ...
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Proving the equivalent condition for primality

Let $p\in\mathbb{N}$, p>1 then p is a prime if and only if for every $a,b\in\mathbb{N}$, p=ab implies a=1 or b=1. if p=2, then a=1 and b=2, but if p=6, then a=2 and b=3 or a=1 and b=6. I ...
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Do mathematicians ever prove that something can or can't be proved?

I was just idly thinking about things people have a hard time proving, like P=NP, etc, and wondering if instead it could be proved that it's provable or unprovable. Is that a thing? Does that ever ...
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Proving that the ball is converx

I need to prove that the ball $B(x,r)=\{y\in \mathbb{R^n}:||y-x||<r\}$ is convex. How to do this?
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Heptagon (septagon) diagonal intersection

Diagonals TV and UW of regular heptagon TUVWXYZ meet at A. Prove that TU+TA=TW (Source: AoPS ItG). My observations: TUVW is an isosceles trapezoid Triangle TUA is congruent to triangle WVA
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Quick proof help dealing with the intersection of sets

For each real number $r \in (1, 3)$, define $A_r$ to be the interval $[0, r)$. Set $B = \bigcap A_r$. Prove that $B = [0, 1]$. I understand this problem up until the part where we have to set $r$ ...
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Prove a function $f$ is one-to-one iff $F^{-1}[f(x)] = x$

Prove a function $f$ is one-to-one iff $F^{-1}[f(x)] = x$ I know that a function is one-to-one if and only if: Let $a$ be in the domain. Then $f(a)=b, b\in$ the range. then the inverse image ...
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proofs using Am-GM inequality

I am getting crazy with this one. Suppose $a_n=(1^2+2^2+3^2+\ldots+n^2)^n$ and $b_n=n^n(n!)^2$. Show that $a_n>b_n$ for all $n$. They suggest to use the Am GM inequality. Thanks!
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Prove that cube connot be tiled with $n>1$ cubes, such that all of them have different side length.

Prove that cube connot be tiled with $n>1$ cubes, such that all of them have different side length. I believe this is not hard problem, but I just do not have an idea how to start. I tried to ...
Proving $T(n) = 1 + \sum_{j=0}^{n-1} T(j)$, $T(0)=1$ implies $T(n)=2^n$
I feel that this is a fundamental question. $$T(n) = 1 + \sum_{j=0}^{n-1} T(j).$$ Given $$T(0) = 1.$$ Show $$T(n) = 2^n.$$ If I substitute values, I can see that the series goes like 1, 2, 4, ...