Does there exist a function f:N→Nf: \mathbf{N} \to \mathbf{N}f:N→N such that f(1)=2f(1) = 2f(1)=2, f(f(n))=f(n)+nf(f(n)) = f(n) + nf(f(n))=f(n)+n for all n∈Nn \in \mathbf{N}n∈N, and f(n)<f(n+1)f(n) < f(n + 1)f(n)<f(n+1) for all n∈Nn \in \mathbf{N}n∈N?