# Countability of a subset of sequences

Let $$\mathcal{A} = \{a \in \{1,2,3,4,5\}^\Bbb N : |a_i- a_{i+1}| = 1 \; \forall i\}.$$ Is the set $$\mathcal{A}$$ countable?

I tried an argument like Cantor's diagonalization process but without success. This problem arises when solving the hiding cat puzzle (https://www.youtube.com/watch?time_continue=2&v=yZyx9gHhRXM). Indeed, if that set is countable and $$\{a^1, a^2,...\}$$ is an enumeration of $$\mathcal{A},$$ then we can define $$a \in \{1,2,3,4,5\}^\Bbb N$$ by $$a_i = a^i_i.$$ Then, the sequence $$a$$ solves the puzzle.

For any $$g\in \{1,3\}^{\Bbb N}$$ let $$g^*\in \{1,2,3\}^{ \Bbb N}$$ where $$g^*(2n)=g(n)$$ and $$g^*(2n-1)=2.$$
Then $$g^*\in A.$$ And if $$g,h \in \{1,3\}^{\Bbb N}$$ with $$g\ne h$$ then $$g^*\ne h^*.$$
So $$\{g^*: g\in \{1,3\}^{\Bbb N}\}$$ is an uncountable subset of $$A$$ because $$\{1,3\}^{\Bbb N}$$ is uncountable (Cantor).
Intuitively, you can encode infinite sequences of $$0$$s and $$1$$s as such a sequence: encode $$0$$ as '$$2,1$$' and $$1$$ as '$$2,3$$'. This ensures that consecutive terms in the sequence are exactly one apart.
For example $$0,1,1,0,1,0,\dots$$ is encoded as $$2,1,2,3,2,3,2,1,2,3,2,1,\dots$$
This defines an injection $$\{0,1\}^{\mathbb{N}} \to \mathcal{A}$$, meaning that $$\mathcal{A}$$ is at least as large as $$\{0,1\}^{\mathbb{N}}$$, which is uncountable.