A question of numbers (one for the maths fans)

The number 0.99999... is defined as an infinite series.If you accept that the number exists, you must accept the concept of infinity.You are of course free to refuse this, but this would have fundemental effects on a lot of other things.When youaccept the notation 0.9999... you accept that the concept is that the 9's are repeated an infinite number of times.Now if you accept this or 0.9999... and 9.9999... then you must accept that
9.9999...-0.9999... =9 is meaningfulInstead of placing nine an infinite numbr of times, you subtract an infinite number of nines.One way to prove that 0.9999... equals 1 is to use infinite series.0.9999... is defined as the sum9/10 + 9/100 + 9/1000 + ...Does this sum converge on a number?The difference from 1 is 0.1 in the first instance, then 0.01, then 0.001 etc, the difference i 1/(10^n) in the n'th instance.Now if 1 does not equal 0.9999... then 1-0.9999... must be some number different than 0. Assume that 1 does not equal 0.9999... , then there must be difference, lets call it d, between them, that is
1-0.9999...=d, where d>0Now we know that the difference between 1 and 0.9999... is 1/(10^n) in after n instances.But we can make n arbitrarely large, since the sum that makes u 0.9999... is an infinete series, but then there will be a number N where 1/(10^N) is smaller than d!But that is contrary to our claim that d>0, so we can conlude that d=0, that is 0.9999...=1!This all hinges on the fact that 0.9999... is a sum of infinite many addends, so if you accept the notation 0.9999..., yuo must accept the conclusion that it equals 1.

not quite right.If you subtract to infinities, you can get 0, any number, or infinity.The natural numbers is a infinite set, he natural numbers and zero is an infinite set.subtract the first from the latter and you end up with 1.Of couse it matters how you subtract. Remove all natural numbers from the rational numbers, and you have an infinte set remaining. But order the rational numbers, indexing so you have a first element, second element, etc, and now, for each natural number, N,remove the N'th element from the rational number and you end up with zero elements.In this case the infinity of rational and natural numbers have the same cardinality. You canno do the same with real numbers and natural numbers, since the real numbers are not discrete.The same converns the algebraic numbers and the transcendant numbers, like PI, they are of a different cardinality.