How is the word "indefinite" or "indefinitely" used in Mathematics? eg, "the terms in the sequence repeat indefinitely"
Let's start with the Cambridge Dictionary definitions of the words, indefinite and indefinitely:
indefinite: not exact, not clear, or without clear limits
indefinitely: for a period of time with no fixed end; for an unlimited or unknown amount of time
For instance, if someone says,
"the terms in the sequence repeat indefinitely for every $n≥n_{0}$ for some fixed $n_{0} \in \mathbb{N}$"
what does he mean by that?
Does he mean,
- "The terms in the sequence repeat at least more than once, though I'm not sure how many times."
- "The terms in the sequence repeat I don't know how many number of times, it could be that they might not repeat at all, i.e $0$ times or once or more than once or infinitely many times."
- "The terms in the sequence repeat infinitely many times."
Now based on the definitions above, indefinitely means "unclear or unlimited", so I'd think 2nd statement seems to be the correct choice. However, often times I feel like people tend to mean "infinitely many times" (unlimited times) when they use indefinitely.
I've come across texts (I forget where), which also seem to imply they mean the 3rd statement, that is "infinitely many times" when they use indefinitely.
So really which is it?
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