>>6
Mathemathics has(excluding non-standard analysis) belief infinite series have a final number as result, not its limit:
0.999...=1(9/10+9/10^2+...=1.0 implying 1-1/10^n=1)
>>10 Btw there is "pathological example" for 0.999...!=1
1. assume 1=0.999... , transform right parts to series
2. 1=9/10+9/10^2+9/10^n+.... ,substracting equals results in 0
3. 1-9/10-9/10^2-9/10^n-...=0 ,transform into series of divisions
4.10/10 -9/10 -> 10/100 -> -9/100-> 1/10 -> 1/10/10/10/10/10...=0
5. 1/10/10/...=1/10^n=0 , now multiply both sides by 10^n
6 1=0*10^n=0 , i.e. 1=0
>>11
Doesn't this disprove >>10 by contradiction? I guess it's like the liar paradox where despite the contradiction effectively refuting the research program mathematicians choose to ignore it. Also does every irrational number have a series which converges to it, probably?