# Three dot ∴ symbol meaning

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.

The symbol $$\therefore$$ means “therefore”.

• i can't accept your answer until 12 mins have passed – Natural Number Guy Mar 3 '19 at 20:38
• That's fine. I can wait. :-) – José Carlos Santos Mar 3 '19 at 20:39

Well, the answer can be found here and here:

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 $$\mathbb N.$$

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