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## Homework Statement:

- Prove .999 repeating = 1

## Relevant Equations:

- .999 repeated = 1

Let there be a number .999 repeated m times where m is a natural number, which is the smallest number less than 1.

Since there is a number .999 + .0000 with a 9 in the m+1 position, it that follows the number of the form .9999, with an infinite amount of 9's, is bigger. Since infinity doesn't end, there is no finite number of the form .9999 with x 9's, x being a natural number, that is less than 1.

Since the number of the form .999 with 9's repeating x amount of times, x being a natural number, is not less than 1, but it's not greater than 1 either, it must be 1.

Since there is a number .999 + .0000 with a 9 in the m+1 position, it that follows the number of the form .9999, with an infinite amount of 9's, is bigger. Since infinity doesn't end, there is no finite number of the form .9999 with x 9's, x being a natural number, that is less than 1.

Since the number of the form .999 with 9's repeating x amount of times, x being a natural number, is not less than 1, but it's not greater than 1 either, it must be 1.

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