Is the supremum of a closed set always in the closed set if this set is bounded? Does it need to be bounded?
Follow Math Help Forum on Facebook and Google+
Originally Posted by inthequestofproofs Is the supremum of a closed set always in the closed set if this set is bounded? Does it need to be bounded? I assume, as your last posts indicated, that you are dealing with closed subsets of . Then the answer is yes. Notice that . But, if is closed then . So, let me ask you: what is ?
(if the supremum exists)
It would be zero, and thus, it is in the closed set
Originally Posted by inthequestofproofs It would be zero, and thus, it is in the closed set
Now prove it
I wanted to ask what about the set , which is a closed subset of .
Then does the supremum of exist?
if so then but would you say that it is in ?
What about the empty set? it is both open and closed ?
what is ? ?
In order to have a supremum, a set must first have an upper bound. does not have an upper bound so no supremum.
View Tag Cloud