Let $f : I = (c,d) \to \mathbb{R}$ be a monotonic function and $-\infty < c < d < +\infty$. How can we show that $\lim_{x \to d} f(x)=\sup\{f(x)\mid x\in{I}\}$ and $\lim_{x \to c} f(x)=\inf\{f(x)\mid x\in{I}\}.$ I think it might be useful to create a sequence with $\lim_{i\to \infty} z_i=c $ (or $d$), but I don't know how to go on.
Show $ \lim_{x \to d} f(x)=\sup\{f(x)|x\in{I}\}$ if $f$ is monotonic on $(c,d)$
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real-analysis
1 Answers
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This is not true unless $f$ is increasing in $(c,d)$. If $f$ is decreasing then $\lim_{x\to d^-}f(x)=\inf f(x)$ and $\lim_{x\to c^+}f(x)=\sup f(x)$.
Proof: Suppose $f$ is bounded and let $a=\sup f((c,d))$ and $\epsilon>0$. Then $\exists x_0\in (c,d)$ so that $f(x_0)+\epsilon>a\Rightarrow -\epsilon
Note: If in addition $f$ is continuous in $(c,d)$ then $f((c,d))=(\lim_{x\to c^+}f(x),\lim_{x\to d^-}f(x))$
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0I see, thanks alot! – 2012-12-16