Exercise 13.
Let \(\phi_n(x) = x^n\) for \(0 \leq x \leq 1\) and show that
\begin{equation*}
\lim_{n \rightarrow \infty} \phi_n(x)
=
\begin{cases}
0, & 0 \leq x \lt 1 \\
1, & x = 1.
\end{cases}
\end{equation*}
This is an example of a sequence of continuous functions that does not converge to a continuous function, which helps explain the need for uniform continuity in the proof of the Existence and Uniqueness Theorem.