Below is the exact quote from textbook Introduction to Set Theory by Karel Hrbacek and Thomas Jech.
4.1 Definition Linearly ordered sets $(A,<)$ and $(B,\prec)$ are similar (have the same order type) if they are isomorphic, i.e., if there is a one-to-one mapping $f$ on $A$ onto $B$ such that for all $a_1,a_2\in A,a_1<a_2$ holds if and only if $f(a_1) \prec f(a_2)$ holds.
The previous results show that there is a rich variety of types of linear orderings on countable sets. It is thus rather surprising to learn that there is a universal linear ordering of countable sets, i.e., such that every countable linearly ordered set is similar to one of its subsets. The rest of this section is devoted to the proof of this important result.
In this textbook, $A$ is countable $\iff$ $A$ is countably infinite.
While I can prove below statement for $\Bbb N$, I'm unable to prove it
i.e., such that every countable linearly ordered set is similar to one of its subsets.
for countable linearly ordered set in general. Please shed me some lights!