Monotonicity and convergence of discretized quartic minimization problems

Consider the discretized problem with discretization parameter $N$, its smallest eigenvalue, and its optimal value, together with the corresponding original problem. Monotonicity and convergence conjecture. The smallest eigenvalue and the optimal value of the discretized problem are monotone nondecreasing with $N$ when $N$ is large enough, and converge to those of the original problem. This conjecture is motivated by numerical results for the discretized problems, which indicate increasing smallest eigenvalues and optimal values as $N$ grows; its general validity is not established in the supplied text.