Earlier quoted context omitted.
Maybe you will find the proof that the infinite series 0.9999... exactly equals 1 interesting: https://en.wikipedia.org/wiki/0.999 ...
Wow, can't believe I've never realised this. How counterintuitive. The 1/3 * 3 argument, I found the most intuitive.
The geometric series proof is less fun but more straightforward.
As a fun side note, the geometric series proof will also tell you that the sum of every nonnegative power of 2 works out to -1, and this is in fact how we represent -1 in computers.