Showing continuous function has min or max using Cauchy limit def.

[CGandC]

Problem: Let $f: \Bbb R \to \Bbb R$ be continuous. It is known that $\lim_{x \to \infty } f(x) = \lim_{x \to -\infty } f(x) = l \in R \cup \{ \pm \infty \}$. Prove that $f$ gets maximum or minimum on $\Bbb R$.

Proof: First we'll regard the case $l = \infty$ ( the case where $l = -\infty$ is similar ). Denote $| f(0) | = M$. By the given there exists $N>0$ large enough s.t. for all $|x| > N$ , $f(x) > M \geq f(0)$. [ The proof continues by using weierstrass theorem, finishes for the infinite case, and then it proves for the case where $l$ is finite ]

My question: I was wondering about how they got to the phrase in red. I know they used the definitions for $\lim_{x \to \infty } f(x) = \infty$ , $\lim_{x \to -\infty } f(x) = \infty$ which are:
$\lim_{x \to \infty } f(x) = \infty \iff$ $\forall M>0 .\exists R_1>0. \forall x \in (R_1,\infty). f(x) > M$
$\lim_{x \to -\infty } f(x) = \infty \iff$ $\forall M<0 .\exists R_2<0. \forall x \in (-\infty,R_2). f(x) > M$

What instantiation they used and how did they choose $N$? ( did they chose $N := max \{ R_1 , |R_2| \}$ after instantiating a variable in the universal quantifiers? ) how from these two definitions I get to the phrase in red?

Thanks in advance for the help and advice!

 

[Gaussian97]

Yes, your choice should work. if you chose $N = \max{\{R_1, -R_2\}}$ and you assume $|x| > N$, knowing that, by definition $N > R_1$, $N > -R_2$ what can you say about $x$ and $R_{1,2}$?

 

[CGandC]

I can say that $x > N > R_1$ or $x< -N < R_2$.
If $x > N > R_1$ then $f(x) > M$
If $x< -N < R_2$ then $f(x) > -|M|$.

How do I get $f(x) > M$ for both cases ( for $x > N > R_1$ or $x< -N < R_2$ )?

 

[mfb]

For $x<-N \leq R_2$ you still have f(x)>M. It's directly coming from the definition of the limit. The limit is positive infinity.

 

[Delta2]

$\lim_{x \to -\infty } f(x) = \infty \iff$ $\forall M<0 .\exists R_2<0. \forall x \in (-\infty,R_2). f(x) > M$

What instantiation they used and how did they choose $N$? ( did they chose $N := max \{ R_1 , |R_2| \}$ after instantiating a variable in the universal quantifiers? ) how from these two definitions I get to the phrase in red?

Thanks in advance for the help and advice!

The above line is wrong. You have to say for all M>0 and the rest remains as it is. Don't forget, even though our independent variable x goes to minus infinity, the limit remains positive infinity so it has to be for all M>0.

 

[CGandC]

Thank you. With this correction things sit perfectly well!

I have one more question - Is the following definition for $\lim\limits_{x \to +\infty} f(x) = -\infty$ correct?:
$\lim\limits_{x \to +\infty} f(x) = -\infty \iff \forall M<0 .\exists R>0. \forall x \in (R,\infty). f(x) < M$

 

[Delta2]

Yes looks fine to me, now the limit is minus infinity so it should be for all $M<0$ (or alternatively M>0 but we require $f(x)<-M$.

 

[CGandC]

Ok thanks for the help! that'd be all.