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For any open interval containing limsup s_n, there exists infinitely many s_n...
Hi all! I'm new to this fantastic forum! Please help me with the following problem! Thanks in advance!
Suppose that the sequence [tex](s_n)[/tex] is bounded in [tex]\mathbb{R}[/tex]. Prove: Given any open interval containing [tex]\limsup s_n[/tex], there exists infinitely many [tex]n[/tex] with [tex]s_n[/tex] inside that interval.
In light of the given statement, what is the issue with "open interval"? That is, does this statement hold (or not hold) if it said "closed interval" instead? And why?
Hi all! I'm new to this fantastic forum! Please help me with the following problem! Thanks in advance!
Homework Statement
Suppose that the sequence [tex](s_n)[/tex] is bounded in [tex]\mathbb{R}[/tex]. Prove: Given any open interval containing [tex]\limsup s_n[/tex], there exists infinitely many [tex]n[/tex] with [tex]s_n[/tex] inside that interval.
Homework Equations
In light of the given statement, what is the issue with "open interval"? That is, does this statement hold (or not hold) if it said "closed interval" instead? And why?