Mathematics

The key insight is recognizing that $0.\overline{9} = 0.999\ldots$ is actually an infinite sum: $\frac{9}{10} + \frac{9}{100} + \frac{9}{1000} + \cdots$. This is a geometric series with first term $a = \frac{9}{10}$ and ratio $r = \frac{1}{10}$, so it converges to $\frac{a}{1-r} = \frac{9/10}{9/10} = 1$. The partial sums $S_n$ get arbitrarily close to 1 as $n$ grows, which is precisely what the limit definition of equality means for infinite decimals.

The key insight is recognizing that $0.\overline{9} = 0.999\ldots$ is actually an infinite sum: $\frac{9}{10} + \frac{9}{100} + \frac{9}{1000} + \cdots$. This is a geometric series with first term $a = \frac{9}{10}$ and ratio $r = \frac{1}{10}$, so it converges to $\frac{a}{1-r} = \frac{9/10}{9/10} = 1$. The partial sums $S_n$ get arbitrarily close to 1 as $n$ grows, which is precisely what the limit definition of equality means for infinite decimals.