/sci/ - Science & Math

Two limit-based proofs that 0.999... = 1:


=====First=====

0.999... is clearly 1 minus the limit of 10^-n as n grows without bound.

|10| > 1.

The limit of any z^-n as n grows without bound where |z| > 1 is by definition 0.

Therefore, 0.999... is 1 - 0, which is 1.


=====Second=====

0.999... is clearly 9 times the limit of the series 10^-1 + ... + 10^-n as n grows without bound.

Let g = 10^-1 + ... + 10^-n
Then g = 10*(10^-2 + ... + 10^-n) + 10^-n
Then g = 10(g - 10^-1) + 10^-n
Then g = 10g - 1 + 10^-n
Then -9g = -1 + 10^-n
Then 9g = 1 - 10^-n
Then g = (1 - 10^-n)/9

Therefore, 0.999... is 9 times the limit of (1 - 10^-n)/9 as n grows without bound.

|10| > 1.

The limit of any z^-n as n grows without bound, where |z| > 1, is by definition 0.

Therefore, 0.999... is 9 times 1/9.

Therefore, 0.999... = 1.

Math is cool and all but at the end of the day all you guys are doing is fucking around with numbers.

That fact that some of these numbers, but not all, correspond to reality in expected ways does not change the fact it's a brutally simplistic and archaic way of viewing the universe. Ff anything, the decimal system is far too limited to probe the fabric of reality. It's just easy to do short division with.

Why devote a lifetime to this fool's errand.