I have some questions about upper limit

[jwqwerty]

upper limit is defined as:
lim sup(Sn) = sup E , where {Sn} is a sequence of real numbers and E is the set of all subsequential limits of {Sn}.

Then if sup E = +∞, why is there a subsequence of {Sn} whose limit is +∞?
Also, if sup E= -∞ is there a subsequence of {Sn} whose limit is -∞?

 

[Mark44]

upper limit is defined as:
lim sup(Sn) = sup E , where {Sn} is a sequence of real numbers and E is the set of all subsequential limits of {Sn}.

Then if sup E = +∞, why is there a subsequence of {Sn} whose limit is +∞?

From the definition. Slightly restating what you wrote, E is the set of limits of subsequences of {S_n}. This means that some subseqence has a limit of +∞.

Also, if sup E= -∞ is there a subsequence of {Sn} whose limit is -∞?

If sup E = -∞, then every subsequence must have this as a limit. Think about it this way: if the largest value in some set is -∞, there's no way to have any smaller (i.e., more negative) values for limits.