>>8809024

The summary (lightly simplified, but not much) of it is this:

Consider the "infinite sum" 0.5 + 0.25 + 0.125 + 0.0625 + 0.03125 + ...; in other words, 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ... . I claim that these infinitely many values together sum to 1.

What that means is the following: consider the sequence of *partial sums* of this infinite sum: the first number, the sum of the first two numbers, the sum of the first three numbers, etc. This yields 0.5, 0.75, 0.875, 0.9375, 0.96875 ...; AKA 1/2, 3/4, 7/8, 15/16, 31/32, ... . This is an infinite sequence of numbers that get ever closer to 1, but never reach it.

Now it so happens that the numbers in this sequence, while all being less than 1, get *as close to 1* as you like. All numbers from 15/16 (= 0.9375) onwards are between 0.9 and 1; all numbers from 127/128 (= 0.9921875) onwards are between 0.99 and 1. More generally, for *any distance D larger than zero* that you can name -- here I named 0.1 and 0.01 -- there is a *point in the sequence* such that *all points afterwards* are no further than D removed from 1. All points in the sequence after 15/16 are AT MOST 0.1 removed from 1 (i.e. between 0.9 and 1.1); all points after 127/128 are at most 0.01 removed (between 0.99 and 1.01).

Based on this, in calculus, we say that the infinite sum has the value 1. For you can make *finite approximations of the infinite sum* that are *as close to 1 as you would like*. This is called a limit, and the formal statement would be that the limit as i goes to infinity of (1 - 1/(2^i)) is 1. Moreover, formally, the SUM for i from 0 to infinity of 1/(2^i) is 1.

0.999..., AKA 0.9 + 0.09 + 0.009 + ..., AKA the sum for i from 0 to infinity of 0.9 * 0.1^i, has a value of 1 by much the same reasoning.

See https://en.wikipedia.org/wiki/Limit_of_a_sequence and https://en.wikipedia.org/wiki/Series_%28mathematics%29 for further introductory reading.