Monotone divergent function diverges to infinity

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 $x \to \infty$.

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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

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$

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Thank you. I am sorry for the late reply. – Chris kim Jul 27 '13 at 2:06