# Two definitions for the least upper bound

The definition for the least upper bound seems to be given in two nearly identical ways, and I wanted to show that these definitions are, in fact, equivalent, so here is a little note.

Consider the following three statements regarding some set of real numbers , and real numbers .

- for all .
- Let for all ; then .
- Let ; then there is some such that .

My impression is that most books use (1) and (2) above to define
as the least upper bound
of . (To me, at least, this seems
most intuitive, since we actually “see” that out of all the upper bounds
of , is indeed the least.) However, some
books e.g. Ross in *Elementary Analysis* (p 21; it’s actually not
part of the real definition, but (1) and (3) are the only “formal” parts
given) seem to use (1) and (3) instead.

Our task, then, is to show that

- (1) and (2) together imply (3), and that
- (1) and (3) together imply (2).

*Proof*. To show the first implication, suppose (1) and (2)
are true. We want to show (3). Let . We need to find such that . Suppose for the sake of
contradiction that such does not
exist, i.e., for all we have
. Then by (2) we have . But , which is a contradiction.

Now suppose (1) and (3) are true; we would like to show (2). Let for all . We now need . Suppose for the sake of contradiction that , i.e., . Then, by (3), there is some such that . This contradicts the above, where was an upper bound for .