Q:

Proving a lim sup inequality in Chapter 12 of Kenneth Ross's "Elementary Analysis"

I am struggling to digest and understand Theorem 12.2 (p. 76) in Elementary Analysis: The Theory of Calculus by Kenneth Ross

Theorem 12.2
Let $s_n$ be any sequence of nonzero real numbers. Then we have
lim sup $\mid$$s_n$$\mid$$^($$^1$$^/$$^n$$^)$ $\leq$ lim sup $\mid$$s_n$$_+$$_1$/$s_n$$\mid$

In proving this, Ross suggests the following:

Let $\alpha$ =  lim sup $\mid$$s_n$$\mid$$^($$^1$$^/$$^n$$^)$ and L = lim sup $\mid$$s_n$$_+$$_1$/$s_n$$\mid$. We need to prove that $\alpha$ $\leq$ L. This is obvious if L = + $\infty$, so we assume that L < + $\infty$. To prove $\alpha$ $\leq$ L it suffices to show:
$\alpha$ $\leq$ $\beta$ for any $\beta$> L

I'm curious as to why/how it suffices to show that $\alpha$ $\leq$ $\beta$ for any $\beta$ > L . How does this condition prevent $\alpha$ = $\beta$> L?
Thank you for your time and effort!

A:

The reason is the following:

$$\alpha \leq \beta \iff \text{ for each }\epsilon >0  \text{ we have } \alpha< \beta+\epsilon$$

You can see with this is true, intuitively, by drawing the situation in the number line. This should help. One can of course give a proof, of the seemingly more obivous

$$x\leq 0\iff  \text{ for each }\epsilon >0  \text{ we have } x<\epsilon$$

P If $x>0$, $x>x/2>0$ is true, with $\epsilon =x/2$. If for some $\epsilon>0$, $x\geq \epsilon >0$ then it follows $x>0$. Note we proved the contrapositives.
Now, since any number greater than $\alpha$ may be written as $\alpha+\epsilon$ for $\epsilon >0$, you get your claim. 

Set $a_n=|s_n|$. I will give you the proof for $\liminf$; and hope you can give that for $\limsup$. Note inequalities must be reversed! For example, you should choose $\alpha >\ell$.
P Set $$\ell =\liminf_{n\to\infty}\frac{a_{n+1}}{a_n}$$
Choose $\alpha <\ell$. By the definition of $\liminf$, there must exist an $N$ such that, for each $n\geq N$, we have that $$\alpha <\frac{a_{n+1}}{a_n}$$
That is, for $k\geq 0$, we have $$ a_{N+k}>\alpha\cdot a_{N+k-1}$$
This gives that $$a_{N+k}>\alpha^k \cdot a_{N}$$
In paricular $n-N\geq 0$ when $n=N+1,\dots$; so 
$$a_{n}>\alpha^{n-N} \cdot a_{N}$$
Now, taking the $n$-th root gives that for $n\geq N+1 $ $$\root n \of {{a_n}}  > \alpha \cdot{\left( {\frac{{{a_N}}}{{{\alpha ^N}}}} \right)^{1/n}}$$
and taking $\liminf\limits_{n\to\infty}$ gives 
$$\liminf\limits_{n\to\infty} \root n \of {{a_n}}  \geq \alpha $$
This means that for each $\alpha <\ell$, 
$$\liminf\limits_{n\to\infty} \root n \of {{a_n}}  \geq \alpha $$
which is saying that  $$\ell \leq \liminf\limits_{n\to\infty} \root n \of {{a_n}}$$
The proof for $\limsup$ is completely analogous.

