Prove that the norm,$\displaystyle {\left\lVert \cdot \right\rVert}$, considered as a function $\displaystyle {\left\lVert \cdot \right\rVert}:\mathfrak{L}^2([-L,L],\mathbb{F})\rightarrow \mathbb{R}$ is continuous. i.e. $\displaystyle f_n\rightarrow f\in{\mathfrak{L}^2([-L,L],\mathbb{F})}$, converges wrt the norm then, $\displaystyle {\left\lVert f_n \right\rVert}\rightarrow {\left\lVert f \right\rVert}\in{\mathbb{R}}$