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## Prove that for every real number $x$, if $x>2$ then there is a real number $y$ such that $y+1/y = x$.

I’m trying to learn about proofs and I’m stuck in Velleman’s book “How to Prove it”. This is the question (ex.6 p.118): Prove that for every real number $x$, if $x > 2$ then there is a real number $y$ such that $y+\frac{1}{y}= x$. I couldn’t do it so I went to see the answer, […]

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## Prove set of matrices with eigenvalue $1$ is closed

The idea of this question is to find a continuous map onto a closed set, then take the preimage of that closed set to show that a particular set is closed. I’m stumped on this one question: Show the set of all $n \times n$ with all eigenvalues equal to $1$ is closed. What kinds […]

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## Is it true that product of rank $n$ matrices is at most rank $n$?

The context is when doing $A^TA$ for a matrix $A \in M_{m,n}(\mathcal{R})$ say. Then $A^TA$ is invertible if column of $A$ are independent (I’m not sure about this fact either)?

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## Symmetric difference – equality proofs.

I have a couple of statements to prove (self-learning, not homework). I’m not able to proceed with any of them (I’ve tried starting from RHS, LHS, etc.). I suppose I’m missing something. I would like to receive some tips or a solution for one of them and I will try to solve the rest. The […]

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## Find the infinite product

Find the infinite product $$\left(\dfrac{2}{1}\cdot\dfrac{2}{3}\cdot\dfrac{4}{3}\cdot\dfrac{4}{5}\cdot\dfrac{6}{5}\cdot\dfrac{6}{7}\cdot\dfrac{8}{7}\cdots\right)$$ I solved it, but my method is not nice one. I solved it using calculus (that’s why tagged). Please give a nice and sweet solution. Answer

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