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

[Anonymous No.8810615]

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.

[Anonymous No.8810616]

your fields medal is in the mail

[Anonymous No.8810620]

Rigorous constructions of the decimals make this nonsense obsolete. Most would put 0.999... as the decimal expansion of 1, where every real number has an unique infinite expansion. That is if it would otherwise be finite its followed by infinite 0's.

[Anonymous No.8810622]

>>8810620
which specific part(s) of the proof is/are you calling nonsense

[Anonymous No.8810626]

>>8810622
The part where you use limits to "prove" 1=0.999... when the latter is undefined except by convention to already be equivalent to 1. This is not a proof it is an attempt to reconcile an inability to understand ambiguity in nonrigorous constructions of decimals.

[Anonymous No.8810632]

>>8810626
0.999... isn't undefined, it's very obviously the limit of the series 9*10^-n as n starts at 1 and grows without bound.

[Anonymous No.8810635]

>>8810632
It is defined. You are assuming things you don't even realize you are assuming. The series you have mentioned is 1 in any well constructed R. To say that series is 0.999... is no different then just claiming it is 1 out of the blue.

[Anonymous No.8810646]

>>8810635
No, there's a difference actually.

0.999... is 0 with a . after it and then infinitely many 9's.

A digit d in any position i after the ., counting from the left, is d*10^-i. A digit d in any position i before the ., counting from the right, is d*10^i. The sum of all such terms is the number represented by that collection of digits. These facts are all intuitively obvious and shouldn't need to be stated; they are SO obvious, that anything which directly follows from them should be nearly as obvious.

The claim that the series is 0.999... directly follows from them.

[Anonymous No.8810648]

Lets make this simple. A finite decimal is defined almost everywhere as a partial sum of smaller powers of 10. Finite decimals are not reconcilable with infinite decimals, because it implies that the decimal representations are not unique. 3.25=3.250000...=3.249999... etc. Solutions have been presented by many people. In Rudin, you would construct a limiting sequence so that each finite subsequence would be smaller than the actual number, but such that the selected integers are less than or equal to their partial sum of lower powers of 10 components in the normal expansion. By this convention, 0.999... is not the expansion of any real number, and 1=1.000.... and so forth.

[Anonymous No.8810653]

>>8810646
Intuition is not a citation when constructing foundational facts about number systems. I suggest you get a book that teaches you this on page 1. You have failed, delete the thread.

[Anonymous No.8810658]

>>8810648
>Finite decimals are not reconcilable with infinite decimals, because it implies that the decimal representations are not unique.
Who gives a shit. Finite decimals are easily reconcilable with infinite decimals, because infinite decimals are just finite decimals shoved through infinite limits. Decimal representations don't have to be unique any more than representations in any other base have to be unique, consider base negative 2.

[Anonymous No.8810664]

>>8810653
Intuition is the only possible citation when constructing foundational facts about any kind of system which can exist and be meaningful independently from our surroundings. If you use anything but intuition, the facts you're constructing are not foundational, the facts you USED to construct them are foundational. (Or, possibly, the facts used to construct *those*, etc.)

[Anonymous No.8810670]

>>8810653
Beside which, are you telling me it's NOT the case that a digit d at the ith position from the decimal place has place value d*10^-i? Because I have infinitely many counterexamples to that claim.

[Anonymous No.8810726]