Nicely done, I've never seen it proven that way.

I just use the formula for sum of an infinite geometric series:
S = a / (1 - r)
where S is the sum, a is the starting term and r is the common ratio
here, S = 0.9 + 0.09 + 0.009 + 0.0009 ...  so a = 0.9 and r = 0.1 (each term
is 1/10th of the term before it).
now,
0.9999.... =  S = 0.9 / (1 - 0.1) = 0.9/0.9 = 1
So 0.9999...  = 1.

Also we should note, since 0.9999... is repeating, it is rational so it can
by definition be expressed as a quotient of integers.  In this case, that's
simply 1/1.

Jason



----- Original Message -----
From: "Liam O'Hagan" <lohagan@GLI.COM.AU>
To: <PICLIST@MITVMA.MIT.EDU>
Sent: Tuesday, May 18, 2004 7:25 PM
Subject: Re: [EE]: Light bulb make square to sine wave


> An old proof of this from high school...
>
> X = 0.99999999 (repeating ad inifinitum)
>
> 10x = 9.999999999999 repeating
>
> 10x - x = 9x = 9.9999999 - .999999 = 9
>
> 9x = 9
>
> X = 1

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