Re: Does the number of nines increase?

On 6/30/2024 10:51 AM, WM wrote:

Le 30/06/2024 à 12:29, Jim Burns a écrit :

On 6/29/2024 2:14 PM, WM wrote:

If you think straight,
then only one conclusion follows:
0.999... < 1.

>
The values of infinite.length decimals
are assigned by a different method from how
the values of finite.length decimals are assigned.
0.999...  ≠  0.9  <  1
0.999...  ≠  0.99  <  1
0.999...  ≠  0.999  <  1
0.999...  ≠  0.9999  <  1
0.999...  ≠  0.99999  <  1
...

>
If you use only definable length,
then always ℵo terms are missing.

Each nonempty set of terms holds a least member.
Each term t has
a last.before term 10⋅t-9  and
a first.after term (9+t)/10
except the first term 0 which has only a first.after.
No terms are incorrectly described by that.
It is complete.

All finite indices guarantee finite length.
0.999...  =  0.999...  <  1

No.
1 is near almost.all (all.but.finitely.many) of
  0.9  0.99  0.999  0.9999  0.99999  ...
for any sense > 0 of 'near'
For that reason,
we assign 1 to 0.999...
The values of infinite.length decimals
are assigned by a method different from how
the values of finite.length decimals are assigned.
0.999...  =  1

0.999...  =  1

>
That is wrong.
If you insert parentheses, nothing changes
>
..((((0,9)9)9)9)... = 0,999...
0.9 + 0.09 + 0.009 + ...
contains ℵo terms with ℵo nines,
all together smaller than 1.

β is the least.upper.bound of the terms
  0.9  0.99  0.999  0.9999  0.99999  ...
least.upper.bound β is not less than 1
| Assume otherwise.
| Assume least.upper.bound β < 1
|
| 10⋅β-9 < β < (9+β)/10
|
| (9+β)/10 is an upper.bound of terms
|
| 10⋅β-9 isn't an upper.bound of terms
| term tₓ exists:
| tₓ > 10⋅β-9
| (99+tₓ)/100 > (99+(10⋅β-9))/100
| tₓ₊₂ > (9+β)/10
| (9+β)/10 isn't an upper.bound of terms
|
| However,
| (9+β)/10 is an upper.bound of terms
| Contradiction.
Therefore,
least.upper.bound β is not less than 1
Also.
least.upper.bound β not greater than 1
least.upper.bound β = 1
And
1 is near almost.all (all.but.finitely.many) of
  0.9  0.99  0.999  0.9999  0.99999  ...
for any sense > 0 of 'near'
Consider sense of 'near' ε > 0
1-ε is not an upper.bound of terms
term tₑ exists: 1-ε < tₑ < 1
finitely.many terms < tₑ
| Assume otherwise.
| Assume infinitely.many terms < tₑ
|
|  0.9  0.99  0.999  0.9999  0.99999  ...
| is well.ordered.
|
| A FIRST infinitely.preceded term tₓ exists
| with last.before.tₓ tₓ₋₁ finitely.preceded.
|
| However,
| if tₓ₋₁ is finitely.preceded
| then first.after.tₓ₋₁ tₓ is finitely.preceded
| Contradiction.
Therefore,
finitely.many terms < tₑ
And
finitely.many terms < ε
And
we assign 1 to 0.999...
The values of infinite.length decimals
are assigned by a method different from how
the values of finite.length decimals are assigned.