I have read the proofs about why $0.9999.... = 1$, which are satisfying. But I can't get the following argument out of my head.
Defining $0.9999....$ : Lets construct a non-terminating but recurring real number n such that all digits before decimal point are zero and all digits after decimal point are 9. Comparing $1.0000$ with $0.99999...$
Digit at ones place in $1.0$ (i.e. 1) $\ne$ Digit at ones place in $0.99999$ (i.e. 0)
Digit at tenths place in $1.0$ (i.e. 0) $\ne$ Digit at tenths place in $0.99999$ (i.e. 9). And so on....
Hence, $1.0 =0.9999...$ does not fit with our original definition of $0.9999...$ Can you find the mistake in the argument (other than saying that in-fact $1.0 = 0.9999...$)? Am I using a incorrect way to define (or perhaps compare) a number (with another)? Please help me. I am new to analysis. Thanks.