I've been asked to find bijection between $\{0,1,2\}^\mathbb{N}$, the set of infinite sequences of $0,1$ and $2$, and the open interval $(0,1)$, and am told that it follows a similar logic to the proof creating a bijection between $10^\mathbb{N}=\{0,1,2,3,4,5,6,7,8,9\}$ and the open interval $(0,1)$.
I can find the bijection between $10^\mathbb{N}=\{0,1,2,3,4,5,6,7,8,9\}$ and $(0,1)$ by picking some $a \in 10^\mathbb{N}$ and showing that it converges to some number $b \in (0,1)$. Noticing that it is possible for two sequences to converge to the same number $b \in (0,1)$, such as $0.5$ and $0.4\bar{9}$, we create $A=$ "set of sequences with two representations" and $B=$ "numbers with 2 representations". I can then show that these two sets have a bijection between them without much difficulty.
What I do not understand is how to transfer this framework once we no longer have the ability to create an infinite sequence corresponding to any number in $(0,1)$ and I am struggling to find a function that allows me to do so. Any tips would be appreciated.
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