And another Functional Equation

Problem :
Determine all functions from the set of positive integers to the set of positive integers such that, for all positive integers and , there exists a non-degenerate triangle with sides of lengths
(A triangle is non-degenerate if its vertices are not collinear.)

Solution :
Lemma 1 -
Let
Now form the sides of a triangle

is periodic
Let and

But this is a contradiction for a big
Hence the lemma.


Lemma 2 - is involutive and thus bijective
From ,
form the sides of a triangle
and thus the lemma.


Main Proof
Let
But

form sides of a triangle.
But since the function is bijective,

Now I will inductively prove
form the sides of a triangle.

But since the function is bijective, all the values less than have been taken, and ,
And hence the inductive argument.

Now again using that the function is involutive,

form the sides of a triangle.
Replacing values from aor inductive argument,