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There are no last end.segments of ℕᶠⁱⁿThat is a contradiction. If there are no common numbers, then all numbers must have been lost. But then no numbers are remaining. Then there are finite endsegments because ∀k ∈ ℕ: |E(k+1)| = |E(k)| - 1.
There are no finitely.sized end segments of ℕᶠⁱⁿ
There are no finite cardinals common to
each end.segment of ℕᶠⁱⁿ
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