A sequence tends to something, a number cannot tend to something.
The sequence
{0.9, 0.99, 0.999, 0.9999,...}
tends to 1.
But 0.9999999999.... does not.
It means infinity is not a real number, not that infinity does not belong in math...
NOOOOOOOOOOOOOOOOOOOO 0.00000000000...0000001 does not exist, it violates the archimedean property of the reals (that for every real x there is an integer n such that nx > 1).
A proof that 0.000000000000....n can't even exist... Actually it's not really a proof, it's an immediate consequence of the Archimedean property of the real numbers.
Also, 0.000000000...n cannot even be defined, since by definition a decimal number of the form
0.x1x2x3x4x5x6....
is...
NO
There is no final digit.
There is no final element in an infinite sequence. You have to let go of your intuitive notions, there is no such thing as a digit that is never "reached".