Question

can you give an example of a linearly ordered set which has all suprema and infima but is not a well-order.

Answer 1

pls help!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Answer 2

It's been a while since I've done ordering, but since no one replied I figure I'd give it a shot:

1. You want to find a set which contains its suprema and infima. The easiest set which guarantees this is a closed and bounded set.

2. You don't want the set to be well ordered, which means there needs to be a subset which has no least element. The easiest example of this that I can think of is 1/n such that n is a natural number.

Combining 1 and 2, define your set to be the interval [-1,1] in the real numbers. I'm fairly sure that'll give you the example you're looking for. However, someone may want to double check my suggestion.