I have trouble understanding the construction in Lighthill's Example 3 on page 16:
The sequence $f_n(x)=e^{-x^2/n^2}$ is regular, in the sense that \begin{align} \lim_{n\to\infty}\int_{-\infty}^\infty f_{n}(x)F(x)dx \end{align} exists for good functions $F$ and is equal to $\int_{-\infty}^\infty F(x)dx$. (Here 'good' can be taken to mean test functions.)
How does one proceed to prove this limit, and is there a general strategy for basic distributions? Clearly, I cannot interchange the limit with the integral, otherwise the result follows trivially and the point of distribution theory, as far as I can tell, is to define distributions via integrals (hence limit is taken last). However, for this very simple case I don't know how to start with reasonable integral estimates without knowing either $F$ or its integral.
Oddly enough, I actually understood Example 6 where he used another sequence $e^{-nx^2}\sqrt{n/\pi}$ to define the Dirac delta distribution, because the integral estimates are very clear. The limit for Dirac delta case also makes it clear why we cannot interchange limit with integral, since the limit does not exist at $x = 0$ (which is the intuitive reason for thinking of delta function as `infinitely peaked') and hence breaks the integral.