THEOREM:
0.999999... = 1
Remark:
all
numerals in this article are to be read in the common
decimal system.
Remark: 0.999999... has the usual meaning of
\(\lim_{N \rightarrow \infty} \sum_{i=1}^{N} 9 \times (0.1)^{i}\).

PROOF
Evaluate 1 / 1
by long division not in the common way: 1 / 1 = 1
remainder 0, but rather as follows:
1.0 / 1 = 0.9 remainder 0.1;
1.00 / 1 = 0.99 remainder 0.01;
1.000 / 1 = 0.999 remainder 0.001;
1.0000 / 1 = 0.9999 remainder 0.0001 etc.
or in the layout as (no longer!?) taught in arithmetic class in
elementary school:
1
) 1.0 0 0 0 . . . ( 0.9999
0.9   
   
1 0  
9  
  
1 0 
9 
 
1 0
9

1

Meaning:
1.0 / 1 = 0.9999 remainder 0.0001

The evaluation of 1.0 / 1 was
halted here after processing four digits beyond the decimal point.
It is clear that the Golden Rule of Division With Remainder: dividend = divisor × quotient +
remainder, is satisfied at each stage of the process.
During any process of long division this rule is an invariant relation
among the quantities involved.
When our specific long division of 1.0
/ 1 is continued indefinitely we observe that the
remainder gets arbitrarily small.
In formula:
$$1
= 1 \times \sum_{i=1}^{N} 9 \times (0.1)^{i} + (0.1)^N,$$where N is
the number of
decimals processed.
When N
increases indefinitely we obtain in the limit 1 =
0.999999... .
END
OF PROOF
REMARK: Among all purely periodic decimal expansions of
positive rationals not
greater than 1, the expansion 0.999999... is the only one which cannot be obtained
by long division performed in the conventional
way, i.e. by subtracting at each stage of the process the greatest
possible multiple of the divisor, thus leaving the smallest possible
remainder. This explains in my opinion part of the mysteries
and
controverses associated with "0.999999... = 1".
This proof can easily be generalised
to show that terminating decimal fractions with digit
d
in the end position (for
instance: 1.35; d
= 5) are equal to their nonterminating counterparts
with the digit d −
1 in that position, which is followed by
999... (1.35 = 1.34999...).
