I was just curious for the proof/theorem for a very close decimal (so when you keep adding a decimal) to equal the next integer such as .999999999999....=1. Thank you.
2026-03-30 09:53:48.1774864428
Proof for how .9999999...=1
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The given number is a geometric series in disguise:
\begin{align*} 0.999\ldots = 9\sum_{n=1}^{\infty}10^{-n} = \frac{9}{10}\left(\frac{1}{1 - \frac{1}{10}}\right) = 1 \end{align*} which converges because its ratio is $0.1 < 1$.