bandlimited convergent graph signals converge in the fourier domain

Let $(W,\mathcal{X})$ be $c$-bandlimited and $(G_n,X_n)$ a sequence converging to $(W,\mathcal{X})$. Then the GFT $({\hat{x}}_n{)}_i$ converges to the WFT ${\hat{x}}_i$

Let $\mathcal{C}$ be the set of eigenvalues in the $c$-bandlimited set. Then for all $i\in \mathcal{C}$, we have

$$\begin{array}{rl}|({\hat{x}}_n{)}_i-{\hat{x}}_i|& =|⟨X_n,{\phi }_i^{(n)}⟩-⟨\mathcal{X},{\phi }_i⟩|\\ & =|⟨X_n,{\phi }_i^{(n)}⟩+⟨\mathcal{X},{\phi }_i^{(n)}⟩-⟨\mathcal{X},{\phi }_i^n⟩-⟨\mathcal{X},{\phi }_i⟩|\\ & \le ||\mathcal{X}-X_n||\cdot ||{\phi }_i^{(n)}||+||\mathcal{X}||\cdot ||{\phi }_i^{(n)}-{\phi }_i||\end{array}$$

By convergence of $X_n$, for all $ϵ>0$, there exists some $n_1$ such that $||X_n-\mathcal{X}||\le \frac{ϵ}{2}$ for all $n>n_1$.

And for $||{\phi }_i^{(n)}-{\phi }_i||$, we have

$$||{\phi }_i^{(n)}-{\phi }_i||\le \frac{\pi }{2}\frac{||T_W-T_{W_n}||}{d({\lambda }_i,{\lambda }_i^{(n)})}\le \frac{\pi }{2}\frac{||T_W-T_{W_n}||}{\underset{i\in \mathcal{C}}{min}d({\lambda }_i,{\lambda }_i^{(n)})}$$

And from $T_{W_n}\to T_W$, for all $ϵ>0$, there is some $n_2$ such that $||{\phi }_i^{(n)}-{\phi }_i||\le \frac{ϵ}{2}$ for all $n>n_2$

Thus, for any $ϵ$ and each $n>max\{n_1,n_2\}$ and for all $i\in \mathcal{C}$, we have

$$|({\hat{x}}_n{)}_i-{\hat{x}}_i|\le ||X_n-\mathcal{X}||\cdot {\cancel{||{\phi }_i^{(n)}||}}^1+||\mathcal{X}||\cdot ||{\phi }_i^{(n)}-{\phi }_i||\le \frac{ϵ}{2}+|\cancel{|\mathcal{X}||}\frac{ϵ}{2\cancel{||\mathcal{X}||}}=ϵ$$

And for $i\notin \mathcal{C}$,

$$⟨{\phi }_i(T_{W_n}),X_n⟩\to ⟨\mathrm{\Psi },\mathcal{X}⟩=0$$

Where $\mathrm{\Psi }\in \perp \text{span}\{{\phi }_i:i\in \mathcal{C}\}$ (orthogonal complement) $◼$