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Some problems keep on coming again and again,which I even fruste to answer.Phew...!
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This problem has security definitely come over a hundred times... if not thousand ;)
+1 just for the first line of your answer as I too feel the same ;)
Easiest question and standardest solution.
There's another way to prove this.
Lets divide this 0.999999999999.... by 3. The resultant would be 0.3333333333..... which is 3 1
3 1 Is a third of a one. Therefore if we m u l t i p l y (0.333333333333..... = 3 1 ) b y 3 , the result would be
0.99999999999999...... = 1
Therefore, the statement made above is || True ||
But you are making here another approximation by stating that a third is equal to 0.333333.... Basically you proved an approximation with another approximation, thus it's not a proof.
if 0.999999999999999... goes on forever and ever it equals one just as 1.999999999999999999999999999... equals 2!
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......I've lost count the number of times I answered this same question. Anyway...
0 . 9 9 9 9 9 9 … = 0 . 9 + 0 . 0 9 + 0 . 0 0 9 + …
This is a sum of a geometric progression to infinity, where
a = 0 . 9 r = 0 . 9 0 . 0 9 = 0 . 1
Therefore,
0 . 9 9 9 9 9 9 … = 1 − r a = 1 − 0 . 1 0 . 9 = 0 . 9 0 . 9 = 1
0 . 9 9 9 9 9 9 … = 1 , therefore the statement is True