Let $^\omega\omega$ be the collection of all functions $f:\omega\to\omega$. We order $^\omega\omega$ lexicographically, that is: For $f\neq g \in \,^\omega\omega$ let $m(f,g):= \min\{n\in\omega:f(n)\neq g(n)\}$, and then we say $f < g$ if and only if $f(m(f,g))<g(m(f,g))$.


This establishes a linear order $\leq\_{\text{lex}}$ on $^\omega\omega$ which can be "inherited" to $S\_\omega$, the collection of all bijections $\varphi:\omega\to\omega$.


**Question.** Is one of $(^\omega\omega, \leq\_{\text{lex}})$ and/or $(S\_\omega, (\leq\_{\text{lex}} \cap \; (S\_\omega \times S\_\omega)))$ order-isomorphic to a subset of $\mathbb{R}$?


**Side note.** I suspect that there is a simple argument establishing that $(^\omega\omega, \leq\_{\text{lex}}) \not \cong (S\_\omega, (\leq\_{\text{lex}} \cap \; (S\_\omega \times S\_\omega)))$, but I couldn't do it right now. I suppose that the "unit vectors" $\{(1, 0, 0,\ldots), (0,1,0,0,\ldots), (0,0,1,0,0,\ldots),\ldots\} \subseteq \, ^\omega\omega$ pose problems when mapped to $S\_\omega$, but I am unable to complete this line of argument.



Embedding $^\omega\omega$ and $S_\omega$ with lexicographic order into $\mathbb{R}$
There's a general argument we can use for any question of this form.


Cantor's theorem shows that every countable linear order embeds into $\mathbb{Q}$. Consequently, every *separable* linear order (= has a countable dense suborder) embeds into $\mathbb{R}$. This gives a positive answer to both your questions; for example, the eventually-zero sequences form a countable dense suborder of $({}^\omega\omega,\le\_\mathsf{lex})$, while the eventually-identity permutations form a countable dense suborder of $S\_\omega$ (although strictly speaking we don't need to think about this separately).


In general, "small" dense suborders are useful for proving embeddability facts; meanwhile, "long" well-ordered suborders are useful for proving non-embeddability facts *(e.g. that there is no order-preserving map from $({}^\omega\omega, \le\_{\mathsf{dom}})$ to $\mathbb{R}$, where $\le\_\mathsf{dom}$ is the partial order of eventual domination)*.


