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MAT425 Real Analysis L8

Prof. Thistleton

Upper and Lower Bounds Some sets of real numbers are finite in extent. That is, we may say that all elements of the set are between two points. Others, for example the set of prime numbers, are unbounded.

Definition 227

We say that the number u is an upper bound for the set S if $s \le u\ \forall s\ \in S$ . If such a number exists we say that S is bounded above. Similarly, the number l is a lower bound for S if $l \le s\ \forall\ s \in S$ . In this case we say that S is bounded below.

Examples:

Definition 239

Let the number u be an upper bound of the set S. We say that u is a least upper bound, LUB of S if there is no upper bound of S less that u. Similarly, the number l is a greatest lower bound, GLB of S if it is a lower bound of S and if no lower bound of S is greater than l.

A least upper bound is also called a supremum (sup) and a greatest lower bound is also called an infimum (inf).

Definition 248

If s = sup(S) and if $s \in S$ then we call s the maximum (max) of S. If $l = inf\ (S)$ and if $l \in S$ then we call l the minimum (min) of S.

Theorem 256

Suppose that, for some number u, it is true that for any $\epsilon \gt 0$ : (i) there is no element of S greater than $u + \epsilon$ and (ii) there is an element of S less than $u - \epsilon$ .Then u = sup(S).