(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Assume the theorem that a continuous bounded function on a closed interval is bounded and attains its bounds.

Prove that if f: R -> R is continuous and tends to +[tex]\infty[/tex] as x tends to +/- [tex]\infty[/tex] then there exists an x_{0}in R such that f(x) [tex]\geq[/tex] f(x_{0}) for all x in R.

2. Relevant equations

3. The attempt at a solution

I think I understand the basic idea behind this but I'm not sure that my proof is rigorous enough.

R is not closed or bounded, however as x tends to +/- infinity f(x) tends to infinity, so won't affect its minimum value (do I need to prove this? If so, how?)

So to consider the minimum value of f we can consider a closed bounded interval [a,b] a,b in R

By the assumed theorem, f is bounded on this interval and attains its bounds, so there exists an x_{0}in [a,b] such that f(x_{0}) = inf f(x) in the interval [a,b]

By definition of an infimum we then know that f(x) [tex]\geq[/tex] f(x_{0}) for all x in R.

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# Continuous bounded function - analysis

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