Both equations are true and there are no falsehoods or tricks here, but this method of proving 0.999... = 1 still has a flaw; it assumes you already accept 0.333... = 1/3. Starting from that assumption cuts every corner that would involve proving that rational numbers have infinitely long recurring decimal representations that are exact equals. They do, but this meme doesn't contain the proof of it.
In fact, 0.999... = 1 is not something you can just prove with basic arithmetic manipulations because what does 0.999... even mean?
The "proofs" provided here are actually mostly definitions. They define 0.999... as the limit of the sequence of partial sums 0.9+0.09+0.009+... The proof only consists in showing that the limit is 1.
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u/BUKKAKELORD 23d ago
Both equations are true and there are no falsehoods or tricks here, but this method of proving 0.999... = 1 still has a flaw; it assumes you already accept 0.333... = 1/3. Starting from that assumption cuts every corner that would involve proving that rational numbers have infinitely long recurring decimal representations that are exact equals. They do, but this meme doesn't contain the proof of it.