Exercise 14

Let $\phi_n(x) = x^n$ for $0 \leq x \leq 1$ and show that $$\lim_{n \rightarrow \infty} \phi_n(x) = \begin{cases} 0, & 0 \leq x \lt 1 \\ 1, & x = 1. \end{cases}$$ 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.