Theorem

sup exists inf exists

Proof:

exists

Now we show that

Show that

1. is a lower bound of .

2. if is a lower bound of -sup

1.

is an upper bound of

is a lower bound of (line added for correction 6:30 a.m. Sep 12)

2. Suppose that is a lower bound of

suppose not

on the other hand

<------(Could someone tell me how this come about.)

Hence, is an upper bound of

By 1. & 2, & =