Consider moving horizontally through a distance of exactly 1 meter. To get to the 1 meter mark, you have to first get to the 1/2 way point. Then, in moving through the last 1/2 meter still left, you have to first get to the half way point. Then, in moving through the last 1/4 meter still left, you have to make it to the half way point. Then, in moving through the last 1/8 meter still left, you have to make it to the half way, point etc., etc. Thus your journey through 1 meter consists of an infinite sequence of steps, each 1/2 the size of the previous one. Adding up all your steps in the 1 meter journey, one gets the result
1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 1/128 + ... = 1
On the left of the = sign is an infinite (never ending) series, whose sum is exactly equal to 1, after all, you really CAN move through a distance of 1 meter! In exactly the same way
.9 + .09 + .009 + .0009 + .00009 + ... = 1
where on the left of the = sign is an infinite (never ending) series. This time your 1 meter journey consists a first step of .9 meters, and then each subsequent step is 1/10'th of the previous one. But the very DEFINTION of .9999... with the 9's never ending is just the sum of the preceding infinite series, that is,
.9999... = .9 + .09 + .009 + .0009 + .00009 + ...
by DEFINITION. Thus
.9999... = 1