What does it mean when you refer to .99999... ? Symbols don't mean anything in particular until you've

In this case the definition is that you're taking the limit of.9 , .99 , .999 , .9999 , etc. What does it mean to say that limit is 1 ? Well, it means that no matter how small a number x you pick, I can show you a point in that sequence such that all further numbers in the sequence are within distance x of 1 . But certainly whatever number you chose your number is bigger than 10−k for some k . So I can just pick my point to be the k th spot in the sequence.

A more intuitive way of explaining the above argument is that the reason.99999...=1 is that their difference is zero. So let's subtract 1.0000...−.99999...=.00000...=0 . That is,

1.0−.9=.1

1.00−.99=.01

1.000−.999=.001 ,

...

1.000...−.99999...=.000...=0

*defined what you mean by them*.In this case the definition is that you're taking the limit of

A more intuitive way of explaining the above argument is that the reason

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