# Prove convergence of whole sequence $f_n$ of solutions to a differential problem to a limit $f$ (without uniqueness assumptions)

let $$\{f_n\}_n \subset C^\infty \cap L^2(\mathbb R^N)$$ be a sequence of functions that solves a linear differential equation $$F_n(f_n, \nabla f_n) = 0$$. Suppose that there exists a subsequence $$n_k$$ such that $$f_{n_k} \to f$$ weakly in $$L^2$$ and that is the weak solution (not necessarily unique) of the limit problem $$F(f,\nabla f) =0$$.

If we had a uniqueness result for the limit problem, it would be trivial that the whole sequence $$f_n \to f$$. What strategy can one use to show this kind of result without relying on the uniqueness for the limit problem?