Example for $\mathbb{N} ^ \mathbb{N}$ linear order

Question

I'm not really sure if something like $ \{1,2,3,...\} < \{2,2,3,...\} < \{3,2,3,...\}$ works.

I don't ask about solution with transitive, antisymmetric and total proof, just for working example. Greetings!

EDIT: Originally the topic was about total order but I had wanted to consider linear order. Sorry for that!

Answer 1

Lexicographic order is a total order on $\mathbb{N}^{\mathbb{N}}$. That is, $$ (a_1,a_2,a_3,\ldots)<(b_1,b_2,b_3,\ldots) $$ if and only if $a_j<b_j$ and $a_i=b_i$ for $i<j$ (which is vacuously true if $j=1$).

Answer 2

Try lexicographic order. (This will be like alphabetical order, but the alphabet is $\mathbb{N}$ and every word is infinitely long.)