- #1
fmam3
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Homework Statement
Directly from the definition, for a sequence [tex](s_n)_{n \in \mathbb{N}} \subseteq \mathbb{R}[/tex] prove that if [tex]x > \limsup s_n [/tex], then [tex]x[/tex] is not the limit of any subsequence of [tex](s_n)[/tex]. (i.e. Do not use the fact that [tex]\limsup s_n[/tex] is the supremum of the set of subsequential limits.)
Homework Equations
I have been told by my instructor that my proof will fail due to problems with inequalities --- but I fail to see where it would fail; i.e. are there any errors where [tex]>[/tex] should be [tex]\ge[/tex] or vice-versa?
The Attempt at a Solution
Please see the attachment.
Thanks all!