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Define the set $$S :=\{x\in\Bbb R: x > 0\}.$$ I know that the infimum of this set is $0$. But what is its minimum? My textbook defines the minimum as an element of the set. Thus it can’t be equal to $0$. So is the minimum non-existent in this case?

Anne Bauval
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Özgür Arslan
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    Nonexistent, correct – BobTheThird Dec 02 '24 at 19:15
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    If you try to define a minimum of that set and call it $m$, I can say, "what about $\frac{m}{2}$?" And so on and so forth. If the minimum is greater than zero, there are infinitely many such numbers, and so you can't have a smallest one based on the definition of that set. – qa test Dec 02 '24 at 19:17
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    More generally speaking, if a set has a minimum, it will be equal to the infimum. Thus, a set has a minimum if and only if its infimum is one of its elements. – Cameron Buie Dec 02 '24 at 20:08

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For all numbers that exist in your set, call $n$ any valid number in your set. Then $\frac{n}{2}$ is always valid in your set also. This property means there can be no minimum value of $n$. (Minimum being the smallest value).

If $n$ is a number in your set, $\frac{n}{2}$ is also a number in your set, and therefore, there is no smallest number (minimum) in the set. Because if $\frac{n}{2}$ is also a valid number, then $\frac{n}{4}$ is also a valid number, etc. There are infinitely many numbers in the set, and no number is the smallest number in the set.

That is a property of the way you defined your set, such that it has no minimum value.

qa test
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