No we're not talking restaurant etiquette, this named result is about the positive integers. It states simply that if X is a non-empty set of positive integers, there is a "least" element of X, that is a value that is less than or equal to every other element in the set.
Seems obvious enough. Depending on how many intermediate results you're willing to establish prior to proving the actual theorem, the proof is pretty straightforward as well. Next question is: why does anybody care?
Well, as with the whole least upper bound thing, it comes in handy in a variety of situations to know that you can actually produce a minimum from any set of positive integers (and, by corollaries, mins and maxes of any bounded set of integers). It's not the sort of thing that you want to have to argue in the middle of a complex proof. So, you name it and get on with life.
A related result is also named: the positive integers are Archimedean Ordered, that is, for every two positive integers, a, b, there is some n such that a < bn. Not exactly sure why that matters, but the immediate corollary is useful when taking limits: for any real number ε > 0, there exists a positive integer n such that 1/n < ε.
And you should never order your steak well. Medium-rare at most.
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