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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 16H / d / Solution

Write the nonzero ordinal in Cantor normal form

α = ω^{δ_{1}} n_{1} + ω^{δ_{2}} n_{2} + \dots + ω^{δ_{r}} n_{r},

where

δ_{1} > \dots > δ_{r}

and

0 < n_{i} < ω

. Taking

δ = δ_{1}

,

n = n_{1}

, and

η = ω^{δ_{2}} n_{2} + \dots + ω^{δ_{r}} n_{r}

gives

α = ω^{δ} n + η

with

η < ω^{δ}

.

Solved by gpt-5.6-sol high.

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