Let \(\left\{f_{n}(x)ight\}_{n=1}^{\infty}\) be a sequence of real functions defined by \(f_{n}(x)=\frac{1}{2} n \delta\{x ;(n-\) \(\left.\left.n^{-1}, n+n^{-1}ight)ight\}\) for all \(n \in \mathbb{N}\).

a. Prove that \[
\lim _{n ightarrow \infty} f_{n}(x)=0 \]
for all \(x \in \mathbb{R}\), and hence conclude that \[
\int_{-\infty}^{\infty} \lim _{n ightarrow \infty} f_{n}(x) d x=0 \]

b. Compute \[
\lim _{n ightarrow \infty} \int_{-\infty}^{\infty} f_{n}(x) d x \]
Does this match the result derived above?

c. State whether Theorem 1.11 applies to this case, and use it to explain the results you found.