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Let x = 0.99...

Then 10x = 9.99...

So 10x - x = 9.99... - 0.99... = 9 [the strings after the decimal point are infinite so they are the same]

ie 9x = 9

So x = 1

Another approach is to say that given any number k, however small, it is possible to find a number n such that 0.999... to n digits is nearer to 1 than k. This is known as the limiting value of 0.99...

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Q: Why is point 99999999 repeated equal to 1?
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Why is 1.999999 repeated not 2?

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