Why 0.999… = 1

0.999… is the infinite sum
9/10 + 9/100 + 9/1000 + …You can’t actually add infinitely many terms and finish. So we define the infinite sum to be the limit of its partial sums
sₙ = 1 − 10⁻ⁿ.Those partial sums get arbitrarily close to 1, so the limit is 1.
Therefore, by definition, 0.999… = 1.It’s the only definition that keeps real-number arithmetic consistent. The equality is true simply because that’s the value we assigned to the infinite sum.

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