Exercise 7.6.15

Question

\begin{enumerate}[label=(\alph*)] 
\item If $c \in C$, what is $\lim_{n \to \infty}f_n(c)$?
\item Why does $\lim_{n \to \infty} f_n(x)$ exist for $x \notin C$?
 \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
\item By construction $f_n(x) = 0$ for $x \in C_n$, and by definition $c \in C \implies c \in C_n \ \forall n $, so $\lim_{n \to \infty} f_n(c) = 0$.
\item If $x \notin C$ then for some $N$, $x \notin C_n \ \forall n \geq N$. Now by construction $f_n = f_N$ over $C_n^c$, so for $n \geq N$, $f_n(x) = f_N(x)$ is constant and therefore $\lim_{n \to \infty} f_n(x) = f_N(x)$.
 \end{enumerate}

Exercise 7.6.15

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Exercise 7.6.15

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