Let's try a naïve inductive proof of 0.999… ≠ 1. Base case: Given 𝑎⁰ = 0, 𝑎⁰ ≠ 1. Inductive case: Given 𝑎ⁿ⁺¹ = 1 - (1 - 𝑎ⁿ) / 10, 𝑎ⁿ ≠ 1 ⇒ 𝑎ⁿ⁺¹ ≠ 1

Let's try a naïve inductive proof of 0.999… ≠ 1. Base case: Given 𝑎⁰ = 0, 𝑎⁰ ≠ 1. Inductive case: Given 𝑎ⁿ⁺¹ = 1 - (1 - 𝑎ⁿ) / 10, 𝑎ⁿ ≠ 1 ⇒ 𝑎ⁿ⁺¹ ≠ 1 This proves that for every 𝑎ⁿ = 0.999…9, there's an 𝑎ⁿ⁺¹ that's a 9 larger and still different than 1, which is similar to your halving the pie example. However, you can see that it always happens that 𝑎ⁿ⁺¹ > 𝑎ⁿ, so the "last" infinite 0.999… is not part of the sequence of the inductive case. My intuitive way to see this is that infinitely repeating decimals are an abomination that breaks the nice property of decimal notation where each number contains a single representation (without zeroes at the beginning or at the end of the decimal). Fractions are the one true way to represent rational numbers.