Suppose that $K$ is a non-empty subset of the set of real numbers and is non compact. Prove that there exists a monotone sequence in $K$ that does not converge to a point in $K$.
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What have you tried? Where are you stuck? Take an example of a non-compact set where this is "not obvious" to you. – Sarvesh Ravichandran Iyer Apr 20 '20 at 15:57
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What is the definition of compactness you're using? In the reals, we can say
$K$ is compact $\iff$ every sequence in $K$ has a subsequence convergent to a limit in $K$
And we know that every real sequence has a monotone subsequence. Combining the two, one finds: $K$ not compact $\implies$ there is a sequence $(x_n)$ in $K$ with no convergent subsequence $\implies$ the monotone subsequence of $(x_n)$ has no limit in K.
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