$L_2$ convergence maintains the sign

Question

Let $X$ a finite dimensional space such that $X\subset H^1.$ Let a sequence of non-negative functions $f_n\in X,\,n\geq1$ and a function $f\in H^1$ such that \begin{equation} \|f_n - f\|_{L_2} \to0,\;\;\;n\to \infty. \end{equation} It is sufficient to conclude to the fact that $f\geq0?$

Answer 1

Yes, $L^{2}$ convergence implies almost everywhere convergence for some subsequence. Hence $f \geq 0$ a.e.