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What is the correct way to answer to this question? I need to define a relation between the minimum of a set and the infimum of this set.

Jack
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    https://en.wikipedia.org/wiki/Infimum_and_supremum –  Nov 07 '19 at 14:36
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    Please read the wikipedia article mentioned above or a similar source. If you have some specific issue you still don't understand afterwards we will gladly help. – quarague Nov 07 '19 at 14:39
  • Possible duplicate of https://math.stackexchange.com/questions/1719659/whats-the-difference-between-a-supremum-and-max-also-infimum-and-min?rq=1 – Era Nov 07 '19 at 15:09

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The minimum, if it exists, will be equal to the infimum. A non-empty subset of $\mathbb{R}$ does not always have a minimum, but will always have an infimum.

For example, consider the interval $(0,1)$ which has no minimum or 'smallest' number since we can always get arbitrarily closer to $0$. But this set does have an infimum: its infimum is $0$. The infimum of a set does not need to belong to the set, whereas the minimum does.

Jack Crawford
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