Suppose I have a sequence $f: \mathbb N \rightarrow \mathbb R$ such that $0 \le f(n) \le 1$ for all $n$. By the Bolzano Weierstrass theorem, that there must be convergent subsequences of $f$. Suppose I collected all the limits of such convergent subsequences into a set $E$. Is it then true that the limit superior of $f$ is the supremum of $E$?
My other question is: For the above sequence $f$, if limit superior of $f$ is positive (strictly), must it contain a convergent subsequence with a positive limit?
If my first question was true, Then if all the limit points in $E$ was equal to $0$ then the supremum of $E$ could be set to $0$.Contradicting the fact that limit superior of $f$ is positive.
