I'm trying to follow an Example (5.1) from a PDE book (Vasy). I was having trouble following the proof of the following question (modified):
Define a bump function: ${\delta_j}^{-n}\phi(\frac{x}{\delta_j})$ where $\phi (x) = e^{\frac{1}{x^2-1}}$ ($x \in [-1,1]$, 0 otherwise) and $\lim_{j\to\infty}{\delta_j} = 0\ \ ({\delta_j} \in \mathbf{R})$. Moreover let $\psi(x)$ be any infinitely differentiable function. Show that if: $$ u_j(\psi) = \int \delta_j^{-n} \phi(\frac{x}{\delta_j})\ \psi(x) \ dx$$ then $$\lim_{j\to\infty} u_j = c\delta_0$$ where $ c = \int\phi(x) \ dx$.
Can anyone help me understand how to show this convergence? I know I want to show that for $\forall \ \epsilon \ \exists \ J$ s.t. $\mid u_j - c\delta_0\mid<\epsilon \ \forall \ j>J$. Then what?
thanks!