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Possible Duplicate:
Real Analysis Boundedness of continuous function

I have a real analysis question that I am having trouble with:

Suppose that $f:\Bbb R\to\Bbb R$ is continuous on $\Bbb R$ and $\lim\limits_{x\to-\infty}f(x) = 0$ and $\lim\limits_{x\to\infty}f(x) = 0$. Prove that $f$ is bounded on $\Bbb R$ and attains either a max or a min on $\Bbb R$.

Should I do this by contradiction, perhaps using the fact that if $f(x)$ is unbounded on $\Bbb R$, then WLOG, suppose that it contains a max. Then there exists $s:=\max f(x)$ where $s>f(x)$ for all $x$ in $\Bbb R$. Then $s>f(x_m)>M$ meaning $f(x_m)$ is bounded on $\Bbb R$.

I just don't think this will work since I also have to prove it has a max or min. I can't just assume it has one.

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    Was one of your infinities supposed to be $-\infty$?2012-10-23
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    Yes, sorry about that2012-10-23
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    My mistake - I missed it when I was reading your post for the first time.2012-10-23
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    Just curious is [this (Elements of Real Analysis - Charles Denlinger)](http://books.google.ca/books?id=5YhXVE9slLsC&pg=PA256&lpg=PA256&dq=%22and+attains+either+a+maximum%22&source=bl&ots=hGvrMPAyn-&sig=4iD5eY7yMwBtJaQtrIdu6z2AVFk&hl=en&sa=X&ei=0PGGUN6PHqTnyAGZk4CABg&ved=0CE4Q6AEwBw#v=onepage&q=%22and%20attains%20either%20a%20maximum%22&f=false) is the source of the problem?2012-10-23
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    Same problem but I am using a different book by Bartle2012-10-23

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