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[] Let S be any non‐empty subset of R. The set of

upper bounds of S is U(S)={uR:ussS}.

lower bounds of S is L(S)={lR:lssS}.

[] Let S be any non‐empty subset of R. Then

the supremum of S in R is the unique smallest element in U(S) .

the infimum of S in R is the unique largest element in L(S) .

the supremum and infimum of S always exist.

the supremum (infimum) is also referred to as the least upper bound (greatest lower bound).

[] Let S be any non‐empty subset of R. Then

a maximum of S in R is an element s*S with s*ssS.

a minimum of S in R is an element s*S with s*ssS.

a maximim and/or minimum of S may not exist.

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