Order-preserving map between $X^*$ and $\mathbb N$ for some finite set $X$

Question

Let $X = \{ 1, \ldots , r \}$ be $r$ symbols with $1 < 2 < \ldots < r$ and define on the set of all finite sequences $X^*$ the lexicographic ordering as given here on Wikipedia. Is it possible to establish an order preserving (i.e. monotone) bijective mapping between $X^*$ and $\mathbb N$?

Answer 1

We've seen two yeses so far. The answer is no. Hint: A natural number has only finitely many predecessors. (Or: there are only finitely many natural numbers between any two natural numbers.)

Answer 2

Hint: You have an ordering on $X$, and if you are familiar with different base expansions of natural numbers, then all you have to do is identify each string in $X^*$ with the number the string represents when interpreted as a base-$r$ expansion of a number. If $r > 1$ is an integer, the base $r$ representation/expansion of a natural number is unique, and in bijective correspondence.