Prove that monotone sequence in $K$ that does not converges to a point in $K$

Question

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 $K$.

Answer 1

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.