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 $(c,d)$
3
$\begingroup$
real-analysis
1 Answers
1
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))$$
-
0I don't see why to set $\delta=d-x_0$, why is the proof not finished at $\left|f(x)-a\right|<\epsilon$? – 2012-12-16
-
0@Rob To be completely typical we need $d-\delta
don't we? – 2012-12-16 -
0Honestly I still can't figure out why this is needed to finish the proof, because $-\epsilon
shows that $\lim_{x\to d}f(x)$=a=sup? – 2012-12-16 -
0To conclude that the limit is $a$ you need $d-\delta
– 2012-12-16 -
0I see, thanks alot! – 2012-12-16