# monotonically increasing sequence

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.

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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.

José Carlos Santos
October 09, 2019 20:31 PM

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.