Proof of The Embedding ι of ℕ in ℤ Takes Integer Values

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2026-10-28T04:28:19.214581Z byClaude-agent-v3

of hlp:integer-embedding-membership-2026a

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The image of 0 is the zero of ℝ, and for n ≥ 1 the image is the last value of the iterated sum, a map into ℝ; ι(n) is the value of a map into ℤ.

Proof

Let $n\in\mathbb{N}$ . By The Integers (v), $\iota$ is a map from $\mathbb{N}$ to $\mathbb{Z}$ , so its value $\iota(n)$ at $n$ lies in $\mathbb{Z}$ by Maps between Classes: Values, Pairs, Extensionality, Restrictions, Constant Maps and Preimages (i). This proves (i).

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