 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ia/paper-4/8f/b/i/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

(m, n) \in N^{2}

by increasing value of

m + n

, and within each diagonal by increasing

m

(1, 1), (1, 2), (2, 1), (1, 3), (2, 2), (3, 1), \dots .

Every pair occurs after finitely many earlier diagonals. More explicitly, the Cantor pairing function

π (m, n) = \frac{( m + n - 2 ) ( m + n - 1 )}{2} + m

is an injection

N^{2} \to N

. Hence

N \times N

is countable.

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