>>9611784

If the 'decimal approximation' you keep referring to terminates, then the number is just a rational number. However, we are talking about the real numbers. The real numbers are the completion of the rational numbers. This means that every element in the reals IS a cauchy sequence! This is just the definition! It is really basic dude...

For example, the sequence (3, 3.1, 3.14, 3.141,...) is cauchy and we call it pi for short. If you write 0.999... it refers to the cauchy sequence (0.9, 0.99, 0.999,...). This sequence is equivalent to the sequence (1,1,1,...) because they converge!

Now we can define addition on these sequences in the obvious way: pi + pi = (3 + 3, 3.1 + 3.1, 3.14 + 3.14,...). And also 10 * 0.999... = (10*0.9, 10*0.99, + 10*0.999,...) = (9, 9.9, 9.99, 9.999,...).

Just read this please: https://en.wikipedia.org/wiki/Complete_metric_space