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Let $X$ be a countably infinite, well-ordered sequence of natural numbers. For each value $i \in X$, append $1$ $i$ times to the beginning of the countably infinite sequence $Y$ of binary digits and then append $0$. For example, $(1,4,3,2,6,3,\overline{0}) \to 1011110111011011111101110\overline{0}$. (Overline denotes an infinitely repeating value.) Inversely, the binary sequence $010110111011110\dots$ would map to $(0,1,2,3,4,\dots)$.

Is this a valid bijection between sequences of binary digits and sequences of natural numbers?

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    Yes it is. You can prove both injectivity and surjectivity separately. – 2017-01-18
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    You can also use the obvious inj from Y to X and use [this](https://en.wikipedia.org/wiki/Schr%C3%B6der%E2%80%93Bernstein_theorem) to get a bij. – 2017-02-17

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