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You want to gift your friend 9 dollars. There is a gift tax of 10% ( = 0.1). If you gift them 9 dollars, they will only get 8.1 dollars.
So you want to gift them an amount such that after tax, it's 9 dollars. How much should you gift?
You could say "okay I'll give 9 dollars, plus its tax of 0.9 dollars, so the gift is 9.9". But then your friend will end up with 8.91. You say "okay, I'll add the tax of the tax, which is 0.09, so the gift is 9.99", but your friend will end up with 8.991. You keep doing this, and you conclude that you need to gift your friend 9.9999... dollars.
Here's a different way to approach the problem: If the gift is taxed 10%, then what your friend keeps is 90%. So you want Gift*0.9 = 9, and so clearly you should gift 10 dollars.
Both approaches are correct, so 9.999... = 10. Subtracting 9 (or dividing by 10) gives that 0.999... = 1.
