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Problem 5

Let N={1,2,3,…}\mathbb{N}=\{1,2,3,\ldots\}. Determine whether there exists a strictly increasing function f:N↦Nf:\mathbb{N}\mapsto\mathbb{N} such that (i) f(1)=2f(1)=2; (ii) f(f(n))=f(n)+nf(f(n))=f(n)+n, (n∈N)(n\in\mathbb{N}).
Step 4 of 5: Verify the functional equation
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In plain words

The golden-ratio floor map turns composition into addition exactly, not approximately.

f(f(n))=f(n)+nf(f(n))=f(n)+n
Detailed analysis

If m=f(n) and delta is the fractional part from the previous step, direct expansion gives the argument of the second floor as m+n plus a number strictly between 0 and 1. Its floor is therefore m+n, which is f(n)+n.