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It means that a set does have elements, and is not an empty set.

Should I write down the definitions here?

Is it " If it were" or " If it weren't"

I didn't get you ...

If a subset of the real number, \( A \subset \mathbb{R} \), were empty, then it would have no sup.

the least upper bound is the supremum. Different words, same thing.
Likewise for glb and inf

I am. The lub = least upper bound. The lub is the supremum. Same thing.

And why is it absent in the definition of upperbound

upper bound or least upper bound?

Just upper bound

My point is why not say "let S be a non empty set of real numbers"

Sorry my computer crashed, and it got restarted.

Okay,... I am getting you....
Thanks.

**we [not well]

what did you mean by that, I am sorry I didn't understand that...