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Whats the meaning of this symbol? Its a three dot symbol: ∴ I read a book, im could not find any definition of this symbol.

This is about continuum property of the natural numbers and the archimedean property:

for some $n\in\mathbb{N}$,

$n>B-1$

$n+1>B$

this should be a proof on the set $\mathbb{N}$ of natural numbers is unbounden above. But I do not understand it.

An answer on how the three-dot symbol is what I am out after. Additional explanation of the proof would be nice to know as well, but not needed.

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3 Answers 3

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The symbol $\therefore$ means “therefore”.

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The three dot symbol $\therefore$ means therefore.

Less common, $\;\because$ means because.

Assume $n > B-1$. By a well-known property of $>$ (add $1$ to both sides), $\;\therefore n+1>B.$

For any potential upper bound $B$ of $\mathbb N,$ by the Archimedean property there is $n \in \mathbb N$ such that $n \ge B > B-1.$ But then $n+1\in \mathbb N $ and $n+1>B,$ so $B$ is not really an upper bound of $\mathbb N$. This contradiction shows there is not an upper bound $B$ of $\mathbb N.$

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This triple dot symbol $\therefore$ is denoted "thus" or "therefore".

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