1. **State the problem:** We want to prove that the repeating decimal $0.999999\ldots$ is equal to $1$. 2. **Use a variable to represent the repeating decimal:** Let $x = 0.999999\ldots$. 3. **Multiply both sides by 10:** $$10x = 9.999999\ldots$$ 4. **Subtract the original equation from this new equation:** $$10x - x = 9.999999\ldots - 0.999999\ldots$$ $$9x = 9$$ 5. **Solve for $x$:** $$x = \frac{9}{9} = 1$$ 6. **Conclusion:** Since $x$ was defined as $0.999999\ldots$, we have shown that $$0.999999\ldots = 1$$ This is a well-known fact in mathematics that the infinite repeating decimal $0.999999\ldots$ is exactly equal to $1$, not just approximately equal.