by Black
Last Updated October 09, 2019 21:20 PM

If $x_n$ is a sequence in reals such that $\lim x_n = +\infty$, then a convergent subsequence may or may not exist.

But what if $\lim x_n = +\infty$ and $x_n$ is monotonically increasing. Is that sufficient condition to say a convergent subsequence does not exist?

EDIT: I meant "$x_n$ is unbounded" instead of "$\lim x_n = +\infty$" everywhere above.

If $\lim_{n\to\infty}x_n=\infty$ then the limit of any subsequence of $(x_n)_{n\in\mathbb N}$ is also $\infty$ and therefore it doesn't converge.

Yes, it is sufficient condition to say that a convergent subsequence does not exist. The reason for that is if the sequence is diverging to infinity, then for any large number $M$ eventually all the terms are going to be bigger than $M$ so you can not have a convergent subsequence.

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