Least upper bound

I am confused about the concept of "least upper bound". Is this line the limt of the {an} sequence. If so, how can we prove it?

the least upper bound of a set E is a number greater than every element of E s.t. if another number is less than the least upper bound, it is in E.

The term least upper bound is a loaded term.
First of all it is an upper bound of the sequence {an}, meaning it is greater than every term of the sequence. More generally the upper bound of some set is some number greater than any number in the set.
Secondly it is the smallest such upper bound. So if A is a least upper bound for {an} and B is some other upper bound then A<B

the least upper bound of a set E is a number greater than every element of E s.t. if another number is less than the least upper bound, it is in E.
This is false, consider the set (0,1) then 1 is obviously the least upper bound, and also 0 is less than 1, but 0 is not in the set.

With a search on google and with your help I understood the point. Thanks for your help guys.

ssd
Think of an upper bound of nevative real numbers. Obviously 0 and any number greater than 0 is an upper bound. But the least is 0.