Exercise 8.2.5

Question

\begin{enumerate}[label=(\alph*)] 
\item Consider $\mathbf{R}^2$ with the discrete metric $\rho(x,y)$ examined in Exercise 8.2.3. What do Cauchy sequences look like in this space? Is $\mathbf{R}^2$ complete with respect to this metric?
\item Show that $C[0,1]$ is complete with respect to the metric in Exercise 8.2.2 (a).
\item Define $C^1[0,1]$ to be the collection of differentiable functions on $[0,1]$ whose derivatives are also continuous. Is $C^1[0,1]$ complete with respect to the metric defined in Exercise 8.2.2 (a)?
 \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
\item In order to reach any $\epsilon < 1$, after a certain point in any Cauchy sequence, all elements need to be identical, with the sequence converging to this identical value. Therefore $\mathbf{R}^2$ (and any set) is complete with respect to the discrete metric.
\item By Theorem 6.2.5 (Cauchy Criterion for Uniform Convergence), any $(f_n)$ which is a Cauchy sequence, uniformly converges to some function $f$. Since uniform convergence preserves continuity, $f$ is also continuous and thus in $C[0,1]$.
\item Recall that Theorem 6.2.5 is an if-and-only-if statement, so any uniformly convergent sequence of functions is a Cauchy sequence. With that in mind, Exercise 6.3.2 is an example of a sequence of functions which converges uniformly to a function which is not differentiable. Thus $C^1[0,1]$ is not complete with respect to this metric.
 \end{enumerate}

Exercise 8.2.5

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Exercise 8.2.5

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