Math 6420 • PDE

\begin{enumerate}
\item[(a)] Define Cauchy sequence, and what it means for a linear space to be complete.

\item[(b)] Let \(C[0,1]\) be the set of real-valued functions which are continuous on \([0,1]\).
Show that \(C[0,1]\) is complete under under the uniform norm
\(\lVert f\rVert=\sup_{x\in[0,1]}|f(x)|\).

\item[(c)] Let \(f_n(x)=x^n\) for \(x\in[0,1]\). Find \(f=\lim_{n\to\infty}f_n\). Is \(f(x)\)
continuous? Does this violate the completeness you showed in (a)? Explain.
\end{enumerate}