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been trying to solve this for some time now.

f is continuous in $ [0,\infty), $ and $\lim_{x\to \infty}f(x) = L . $ prove that if there exist $x \ge 0 $ such that f(x) < L then f has a minimum in $ [0,\infty)$.

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  • $\begingroup$ Are you trying to show that $f$ has a local minimum or a global minimum? $\endgroup$ – Jim H Sep 10 '17 at 13:01
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There exists some $y\in[0,\infty)$ such that $f(t)>f(x)$ for $t\ge y$. Then $f$ meets a minimum in $[0,y]$.

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  • $\begingroup$ @anderstood Thank you, I have remade my answer. $\endgroup$ – ajotatxe Sep 10 '17 at 13:10

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