0.(1)

_{10}= (0.1)

_{9}

(0.0)

_{9}+ (0.1)

_{9}= (0.1)

_{9}

(0.1)

_{9}+ (0.1)

_{9}= (0.2)

_{9}

(0.2)

_{9}+ (0.1)

_{9}= (0.3)

_{9}

(0.3)

_{9}+ (0.1)

_{9}= (0.4)

_{9}

(0.4)

_{9}+ (0.1)

_{9}= (0.5)

_{9}

(0.5)

_{9}+ (0.1)

_{9}= (0.6)

_{9}

(0.6)

_{9}+ (0.1)

_{9}= (0.7)

_{9}

(0.7)

_{9}+ (0.1)

_{9}= (0.8)

_{9}

Finally,

(0.8)

_{9}+ (0.1)

_{9}= ?

Well... in any numeration base, when you finished the digits you restart at 0 and add 1 to the next order, so:

(0.8)

_{9}+ (0.1)

_{9}= (1.0)

_{9}

Recasting the above in base 10, we have

0.(8) + 0.(1) = 1.0

However, in base 10 we still have a digit left, so it is equally valid to write

0.(8) + 0.(1) = 0.(9)

just as it is valid to write 0.(7) + 0.(1) = 0.(8).

Given that the left side is identical, it follows that 1 = 0.(9). QED.

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