In this short, we use a geometric representation and the Archimedean principle to show why 0.99999... equals 1. This fact tends to be a bit controversial in the mathematics classroom, but the only way to make sense of an infinite sum is to treat it as a limit. Thus, we can use the term equality when discussing infinite sums like 0.999... and there is no number this sum could equal besides 1.

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To see a longer video with a similar fact (and perhaps more justification at the end), see
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Here is a playlist with other geometric sums dissection proofs:
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