Why does 0.999... equal 1?

[chickenguy]

Need urgent help

sorrypeople if this annoys u but please don't close this thread. i need urhent help with my 0.999...=1 project and i have a new thing my teacher told me would help, but i don't understand!
0.9=1- 1/10(fraction)
0.99=1-1/100(fraction)
0.999=1- 1/1000(fraction)
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1-(approaches zero)=1

sequence and series.
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Also, any background information and/or history and/or opinions and/or information would be very greatly appreciated!1

 

[matt grime]

OPinions are not required. The real numbers are by construction somewhere where convergent sequences of rationals have unique limits. the sequence 0.99..9 with n nines converges to 1, that is what the above states - the distance from it to 1 is 10^{-n} which converges to zero.

so 0.999... the infinite string of nines which, by definition also represents the limit of that sequence mustbe equal to one by fiat.

 

[joeboo]

Let x = 0.999999...
10x = 9.999...=9+0.999...=9+x
10x - x = 9
9x = 9
x=1

0.999... = 1