The most effective proof that there exists a non-computable function from N to N

For n∈N, f(n) denotes the smallest b∈N such that if a system of equations S⊆{1=x_k, x_i+x_j=x_k, x_i·x_j=x_k: i,j,k∈{0,...,n}} has a solution in N^{n+1}, then S has a solution in {0,...,b}^{n+1}.  The author proved earlier that the function f:N→N is computable in the limit and eventually dominates every computable function g:N→N. We present a simple code in MuPAD which for n∈N prints the sequence {f_i(n)}_{i=0}^∞ of non-negative integers converging to f(n). For n∈N, h(n) denotes the smallest b∈N such that if a system of equations S⊆{x_i+1=x_k, x_i·x_j=x_k: i,j,k∈{0,...,n}} has a solution in N^{n+1}, then S has a solution in {0,...,b}^{n+1}. The function h:N→N is computable in the limit and eventually dominates every computable function g:N→N. A bit shorter code in MuPAD computes h in the limit. It establishes the most effective proof that there exists a non-computable function from N to N.