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I just read in a supplementary textbook that "An infinite set may not have a maximum or minimum, but it will always have a supremum and infimum."

I'm a little perplexed -- is this true? What, for example, is the supremum of the real numbers, or the infimum of the real numbers?

I can imagine that any bounded infinite set has a supremum/infimum, but if a set is unbounded (i.e. $\mathbb{R}$), then how can it have a greatest lower bound or least upper bound?

Many thanks, and forgive my ignorance.

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$$\sup\{r\in\Bbb R\}=+\infty\,\,,\,\,\inf\{r\in\Bbb R\} = -\infty$$ – DonAntonio Oct 20 '12 at 0:53
up vote 4 down vote accepted

Your idea is exactly right, but we often have a convention that a set with no upper bound, such as the positive real numbers, has a supremum of "$\infty$", and a set with no lower bound has an infimum of "$-\infty$". In this sense every set has a supremum and an infimum, although it may not have a minimum or a maximum.

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Brilliant -- thank you so much for the clarification! Makes perfect sense now. – James Evans Oct 20 '12 at 0:57

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