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10/04/11(Tue)18:38 No.20381854Let {a_n} be a series of real numbers defined by a_n = a_(n-1) + 9/(10^n) with a_1 = 9/10.
The first few terms of the sequence are: 9/10, 99/100, 999/1000, ...
Then,
assume, for the sake of a contradiction, that 0.999... is not equal to
1. There must exist a number, c, such that 0.999... + c = 1. In other words, the difference between the two cannot be equal to zero. It may be stated as: |a_n - c| > 0 for all n.
However,
if we try to find this number c, we see that no matter what term in the
sequence we take, there will always be a larger one such that the c we
have chosen does not work. Thus, we arrive to a contradiction as the
number c cannot exist to satisfy the above inequality. |