Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $f:(a,\infty) \to \mathbb{R}$ and $f$ is monotonically increasing.

How to show:

If $f$ diverges as $x \to \infty$ then $f$ must diverge to $\infty$ as $x \to \infty$.

share|cite|improve this question
I think you are missing some conditions. – copper.hat Apr 5 '13 at 7:35
@copper.hat/ I think divergent & monotone increasing imlies diverges to infinity. – Chris kim Apr 5 '13 at 8:09
last version is ok.. – Halil Duru Apr 5 '13 at 8:25
up vote 2 down vote accepted

Well , divergence to infinity as $x\to \infty$ means that :

$\forall M$ $\exists $b such that $x>b$ implies $f(x)>M$.

And you know that given $M$ there is some b with $f(b)>M$.

[Otherwise , we would have boundedness on $[a+\varepsilon,\infty)$ which would imply

convergence as $x\to \infty$ as a result of monotonicity.]

But $f$ being monotonic increasing , we also have $f(x)>M$ for all $x>b$.

So we conclude that $\lim _{x\to\infty} f(x)=\infty $. $\hspace{64mm} $$\blacksquare$

share|cite|improve this answer
Thank you. I am sorry for the late reply. – Chris kim Jul 27 '13 at 2:06

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.