Someone asked this question about how many ways there are to prove $0.999\dots = 1$ and I posted this:
$$ 0.99999 \dots = \sum_{k = 1}^\infty \frac{9}{10^k} = 9 \sum_{k = 1}^\infty \frac{1}{10^k} = 9 \Big ( \frac{1}{1 - \frac{1}{10}} - 1\Big ) = \frac{9}{9} = 1$$
The question was a duplicate so in the end it was closed but before that someone wrote in a comment to the question: "Guys, please stop posting pseudo-proofs on an exact duplicate!" and I got down votes, so I assume this proof is wrong.
Now I would like to know, of course, why this proof is wrong. I have thought about it but somehow I can't seem to find the mistake.
Many thanks for your help. The original can be found here.
Answer
The problem is that you are assuming 1) that multiplication by constants distributes over infinite sums, and 2) the validity of the geometric series formula. Most of the content of the result is in 2), so it doesn't make much sense to me to assume it in order to prove the result. Instead you should prove 2), and if you really want to be precise you should also prove 1).
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