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A comment on Now I’m calculating with constructive reals. Parent comment
1. Analytical
0.9999... = \sum_{n=1} ^{\infty} (9 * 0.1^n) = 0.9/(1 - 0.1) = 0.9/0.9 = 1
The first equality is by definition of strings of the form "0.abcdef...". The second is the sum of geometric series.
2. Algebraical
x = 0.9999... 10x = 9.99999... 9x = 10x - x = 9.9999... - 0.9999... = 9 x = 1
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1. Analytical
0.9999... = \sum_{n=1} ^{\infty} (9 * 0.1^n) = 0.9/(1 - 0.1) = 0.9/0.9 = 1
The first equality is by definition of strings of the form "0.abcdef...". The second is the sum of geometric series.
2. Algebraical