# Tagged Questions

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### Axiom of choice and convergence

Hi fellows, I was wondering. Is the axiom of choice used to show that $\mathbb{R}$ is complete? If yes, is there a way to construct monotonic bounded sequences that do not converge? Thanks in ...
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### Maximum sum of 3 consecutive numbers in a permutation [closed]

Given that $X = \{0, 1, 2, ..., 7, 8, 9\}$, and $P$ is a permutation on $X$. Let $M(P)$ be the maximum sum of 3 consecutive elements. For example, if $P = (0, 2, 4, 1, 5, 7, 9, 3, 8, 6)$, then $M(P)$ ...
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### Are there any mathematical objects that exist but have no concrete examples? [closed]

I am curious as to whether there exists a mathematical object in any field that can be proven to exist but has no concrete examples? I.e., something completely non-constructive. The closest example ...
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### reduced ⊗ reduced = reduced; what about connected?

Several questions actually. All rings and algebras are supposed to be commutative and with $1$ here. (1) Let $k$ be a field, and let $A$ and $B$ be two $k$-algebras. I need a proof that if $A$ and ...