Yes. f(0.999...) is not infinity, it's just undefined. This resolves the paradox.

>>16951291
Nah,
1/(1-0.9) = 10
1/(1-0.99) = 100
1/(1-0.99999999) = 100000000
Etc.
1/(1-0.999...) = +infinity
Not undefined.

>>16951275
f(x) = x^2
f(1) = 1
f(-1) = 1
1 = -1

>>16951307
>1/(1-0.9) = 10
>1/(1-0.99) = 100
>1/(1-0.99999999) = 100000000
Yes, and the limit as x goes up to 1 if 1/(1-x) is positive infinity. But that's not the same as 1/(1-0.999...) .

>>16951307
For f(0.999...) you need to take the limit inside the parentheses, not outside. These are not always interchangeable! Otherwise you can't say "A=B then f(A)=f(B)" since A nolonger represents an actual value if you want the notation to work that way. Taking the limit inside the parentheses you just get f(1) again.

>>16951318
A and B need to be very close like 0.999... and 1.
Here it's obvious that 1=/=-1

>>16951340
>1/(1-0.99999999) = 100000000
>Etc.
Is not the same as :
>1/(1-0.999...) = +infinity
Wut ?

This is a philosophical question.

Do you believe there is a real number between 0.999 and 1?

>>16951275
+infinity comes out of thin air
1/(1-0.999…) = 1/0
You’re embarrassing yourself continuing to repost

>>16951598
1/(1-0.9) = 10
1/(1-0.99) = 100
1/(1-0.99999999) = 100000000
1/(1-0.99999999999999999999999999999999) = 100000000000000000000000000000000

Is it enough for you to say :
1/(1-0.999...) = +infinity

>You’re embarrassing yourself continuing to repost
You're welcome.

>>16951586
>number between 0.999 and 1?
I guess you meant 0.999... and 1.
It depends in what game you'd like to play.
0.AAA...(base 11) is closer to 1 than 0.999...(base 10)
0.BBB...(base 12) is again closer to 1 than 0.AAA...(base 11).
Etc.

>>16951907
>Is it enough for you to say :
>1/(1-0.999...) = +infinity
No, it is not. Consider studying limits do learn how these relations work, and what conclusions you can and cannot draw from a pattern like this.

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>>16951275

>>16951907
1.000... and 0.(b-1)(b-1)(b-1)... are both ways to write the exact integer 1 in any base b.
>0.999... = 0.AAA...(base 11) = 0.BBB...(base 12) = 1

>>16951586
Yes. 0.0001

No paradox. Recurring decimals are just a result of the base you're in.

If we used base 12, a third would be 0.4, and not 0.333..
3 thirds make a whole. Whether it's 3 lots of 0.333.., or 3 lots of 0.4.

Same goes for the 0.999.. situation.

None of this would have happened if Bourbaki had listened to Alex and you know it.