0.9999.... = 1 ?

I saw this quick video the other day with the title "Simple proof for $0.999... = 1$" and thought, "can I do this on my own" without watching. The fortunate result was pretty much yes, but my approach was using the concept of limits and was involved compared to the algebra approach shown in the video after I watched it. So I want to document the proof using the algebraic method in the video so I can remember that there are sometimes more straightforward ways to do the same thing. The idea is that we can start with the variable $x$ which we say is:

$$ x = 0.999\cdots $$

Then if we multiply $x$ by $10$, we have

$$ 10x = 10*0.999\cdots$$

which can be rewritten as

$$ 10x = 9 + 0.999\cdots$$

and substituting $x$ back-in on the right hand side,

$$ \begin{aligned} 10x &= 9 + x \\ 10x - x &= 9 \\ 9x &= 9 \\ \therefore x &= 1 \end{aligned} $$

The one thing that I questioned immediately though, is this is a valid mathematical proof or more of a conceptual demonstration of infinite decimal numbers. It most certainly looks to be the latter as discussed on the Wikipedia page. Since I'm not a professional mathematician I can't say why this algebraic approach is most likely not a rigorous proof, but again seems like a nice reminder that there are different ways to convey relations.

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