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.
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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 $$f(x_0)+\epsilon>a\Rightarrow -\epsilon<f(x_0)-a<a-f(x_0)<\epsilon$$ Since $f$ is increasing $f(x)>f(x_0)$ for $x>x_0$. Thus, $$-\epsilon<f(x)-a<\epsilon\Rightarrow \left|f(x)-a\right|<\epsilon$$ for $x_0<x<d$. Letting $\delta=d-x_0$ completes the proof. If $f$ is unbounded the proof is similar (and simpler) but with $a=+\infty$. 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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