>>132706051. I think infinite decimals is bad notation filled with assumptions that are not always made explicit. My suggestion is to use simpler notation like fractions.
2. Infinite decimal notation assumes that infinity is a valid concept. The ... at the end doesn't mean "as many as you like", it means "infinitely many". If you think infinity is an invalid concept, that's fine, but then that just mean that infinite decimals is an invalid notation.
3. To get an number from decimal notation you add up all numbers associated with each digit times their "positional factor". For example, 123.45 is (1*100) + (2*10) + 3 + (4 * 0.1) + (5 * 0.01).
4. It's the same thing for infinite decimals. The problem is that we can actually add all of the digits one at a time (since there are infinitely many of them). So we take the limit of the partial sums instead. So to calculate 0.999..., you have to calculate the limit of the sequence 0, 0.9, 0.99, 0.999, ... . This limit happens to be 1. 1 is the smallest number bigger than all of the numbers in that sequence.
Note: THIS IS A DEFINITION!!!
There is no "proof" that infinite sums are equal to the limit of their partial sums.
We defined infinite sums as the limit of their partial sums.
Just like there is no "proof" that 2 is the number that comes after 1.
We defined 2 as the number that comes after 1.
You might not like that definition, just like you might not like the definition of "fruit" that considers a tomato to be a fruit.
But it's actually a nice definition with nice properties.
If you add the limits of two sequences, it's going to be equal to the limit of the sequence you get by combining both sequences, so adding infinite decimals just happens to work in the way we expect.
There are other nice properties like this, which make me think this is a useful/insightful definition.
