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## Homework Statement

Give an example of a bounded set that has neither a maximum nor a minimum. (The proof below is given by the book).

We claim that the set ##(0,2)## is bounded and has neither a maximum nor a minimum.

Proof: For each ##x \epsilon (0,2)##, we know that ##0 < x < 2##. Therefore 0 is a lower bound of the set and 2 is an upper bound. Thus, (0,2) is bounded. To see that it has no maximum, suppose to the contrary that ##s## is a maximum of the set ##(0,2)##. Then, by definition of maximum, s must be in the set ##(0,2)##. But

##0 < s < \frac {2+s}{2} < 2## and therefore ##\frac {2+s}{2}## is in the set (0,2) and larger than s, a contradiction. In a similar fashion, you can check that there is no minimum.

## Homework Equations

## The Attempt at a Solution

I don't get where ##\frac {2+s}{2}## comes from. I know that since ##s < 2##, then

##s + 2 < 2 + 2## so ##s + 2 < 4## so ##\frac {s+2}{2} < 2##. But how do we know ## s < \frac {s+2}{2} ##