# 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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### Nature of the mapping $\gamma\colon\mathbb{N}\to\mathbb{N}$ and its solutions.

Start by supposing that $\gamma$ is defined as $$\gamma(n)= \begin{cases} (n+1)/2 & \text{if n is odd},\\ n/2 & \text{if n is even}. \end{cases}$$ Then, for each $n\in\mathbb{N}$, the ...
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### Prove that a function that maps a discrete metric space to any metric space is continuous [closed]

Let $f:D→M$ where $M$ can be any metric space and $D$ is any set with the discrete metric. Prove that $f$ is continuous. I'm not sure where to begin with this.
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### Prove a set is not connected

Let $A$ $\subset$ $M$ such that $A$ ≠ Ø and $A$ ≠ $M$. If the boundary of $A$ = Ø, prove that $M$ is disconnected. I know I'm supposed to find an intersection that is empty in order to show it is ...
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### Prove that if $p$ is a prime number and $p\neq3$, then $3$ divides $p^2+2$

Can someone help me with this? I'm not sure how to approach it.. anything would be helpful! Prove that if $p$ is a prime number and $p\neq 3$, then $3$ divides $p^2+2$ In my textbook the hint it ...
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### Show that such numbers exists

Show that there exists number $x \in (0, \frac{\pi}{2} )$ such that: $$\tan x= \sqrt x$$ How to approach this, because i have utterly no idea.
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### Proving extreme value theorem; is showing maximum enough?

Can we prove the extreme value theorem by merely showing that a maximum exists (if $f$ is continuous and defined on a closed, bounded interval in $\mathbb{R}$) because then we'd apply this "half" of ...
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### Showing there are infinitely many mutually disjoint subsets of $\mathbb{N}$.

Problem: Show that there exists an infinite number of mutually disjoint subsets of the set of natural numbers. My idea: Let $P$ denote the set of all prime numbers. We know $P\subseteq\mathbb{N}$ ...
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### Prove that the second derivative is positive iff the function is convex.

Well, I want to prove the following: Let $f:(a,b)\to\mathbb R$ be double differentiable then $f$ is convex iff $\;\;f''(x)>0$ for all $x\in (a,b)$. Then I tried te following: $\Rightarrow]$ Lets ...
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### Prove that this condition is true on an Zariski open set

Why is there an Zariski open set of $P\in GL(n,\mathbb{R})$ such that $P\,\text{diag}(1,\dots,1,-1)P^{-1}$ can be conjugated by a diagonal matrix $D$ to get an orthogonal matrix? Note that ...
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Let $n>1$ be an integer. Let $M$ be a set of closed intervals. Suppose that the endpoints $u$ and $v$ of each interval $[u,v] \in M$ are natural numbers satisfying $1\le u < v \le n$ and ...
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### Prove that if $9\mid(x^3 + y^3 + z^3)$ then $3\mid xyz$ for integers $x, y, z$

how would I prove that if 9 divides $x^3+y^3+z^3$ then $3$ divides $xyz$? I've thought about it and don't really know where to start. So far, I know that $x^3+y^3+z^3$ is congruent to $0 (\mod 9)$. ...
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### Difference between two relations on the set of real numbers

Let $R$ be a relation on $\mathbb{R}$ such that $xRy$ iff $|x|< 2$ $\textbf{and}$ $y = 3$ and $S$ be relation on $\mathbb{R}$ such that $xSy$ iff $|x|< 2$ $\mathbf{or}$ $y = 3$. Im trying to ...
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### Proof: The reduced row echelon form of a matrix is unique.

If $A \in M_{m\times n}$ with real entries, then there exist a unique matrix $R$ in row echelon form such that $A\sim R$, where $R$ comes from $A$ after performing elementary operations. How can I do ...
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### Show that if $f : \mathbb{R}^{2} \to \mathbb{R}$ continuously differetiable then $f$ is not inyective

Well my question this time is: How to show that $f : \mathbb{R}^{2} \to \mathbb{R}$ continuously differetiable then $f$ is not inyective I was trying to consider the function $g(x,y)=(f(x,y),y)$, ...