Quote:
Originally Posted by ExTexan
I'm not sure anything was proven! :confused
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The mathematical
meaning of the repeating decimal .999... is a
limit.
It is the limit of the
sequence of partial sums of the
infinite series 9/10 + 9/100 + 9/1000 + ...
The sequence of partial sums of the above series is equal to (1-1/10), (1-1/100), (1-1/1000), ... (1-1/10^n)
Using the
definition of limit of a sequence:
Quote:
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A real number L is said to be the limit of the sequence Xn if and only if for every real number ε > 0, there exists a natural number N such that for every n > N we have | Xn−L | < ε.
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It can be shown that the limit of the above sequence is "1".
Therefore, "1" and ".999..." mean exactly the same thing. They are two different ways of writing the
same real number.