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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 5 of 5: Conclude existence
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In plain words

An explicit construction settles the existence question.

f:N→N existsf:\mathbb N\to\mathbb N\text{ exists}
Detailed analysis

The candidate is integer-valued, strictly increasing, has the required first value, and satisfies the required identity. Therefore the answer is yes.