The following is an excerpt from the class notes for my PDEs class
Does anyone know how exactly the Lebesgue Dominated Convergence Theorem could have been applied here? My first instinct is to cry out in agony. Isn't the Lebesgue theorem only applicable in L1? In the actual working of the question from class, this is a complete transcription of what the professor did:
\begin{align} ||\phi_n - u_n||_{L^2}^2 &= ||\phi_n -\phi_n +\eta_n\phi_n||_{L^2}^2 \\ &= \int_{\mathbb{R}^N} \eta_n^2(x)\phi_n(x) dx \\ &= \int_{\mathbb{R}^N} \eta_n^2||\phi_n - \delta_0 + \delta_0||^2 \\ & \leq 2 \int_{\mathbb{R}^N} \eta_n^2(\phi_n - \delta_0)^2 dx + 2\int_{\mathbb{R}^N}\eta_n^2\delta_0^2(x)dx& \\ & \qquad \downarrow \qquad\qquad \qquad \qquad \qquad \downarrow \text{Apply Lebesgue Dom. Con.}\\ & \qquad \ 0 \quad \quad \quad \quad \quad \quad \quad \quad \quad\ \ \ 0 \end{align} I'm completely lost as to trying to parse what the professor did, to the point where I'm not sure if I'm lost, or if he has made a mistake somewhere?
Edit: In the class notes, $\eta_n$ was given as a $\left( \overline{B(0,\frac{1}{n})},B(0,\frac{2}{n})\right)$ cutoff function