Explanation needed on a sequence proof

The Theorem is every convergent sequence is bounded:

$\displaystyle (a_n)\rightarrow\alpha$ Then, by Theorem 2.1 (in my book), $\displaystyle (|a_n|)\rightarrow |\alpha|$.

Quite arbitrary, let us choose $\displaystyle \epsilon =1$ (so I could pick any number not just 1?), and then use the definition of a limit to conclude there exists N such that $\displaystyle \left| |a_n|-|\alpha|\right |< 1, \ \forall n>N$.

That is for n > N,

$\displaystyle |\alpha|-1<|a_n|<|\alpha|+1$.

(I don't understand what is going with the max part)

Hence, $\displaystyle \forall n\geq 1$

$\displaystyle |a_n| \leq \text{max}\{|a_1|,|a_2|,\cdots , |a_N|, |\alpha|+1\}$

and so $\displaystyle (a_n)$ is bounded.