I am having trouble with the following problem.
Suppose $C = \{x ∈ V : \Vert x \Vert = 2\}$, a subset of the normed vector space ($V,\Vert \Vert$). Show that if V is complete then C is complete.
Possible Answer: Suppose that $C$ is a closed subset of $V$, and let $\{x_n\}$ be a Cauchy sequence in $C$. Since $V$ is complete there is some $L\in V$ such that $L$ is the limit of $\{x_n\}$.
We need to show that $x\in C$. By the definition of convergence for every $\epsilon>0$ there is some $N$ such that for all $n>N$, $\|L-x_n\|<\epsilon$. Therefore for every $\epsilon>0$ we can find other some point $x_0$ from $S$ such that $\|L-x_0\|<\varepsilon$. Since $C$ is closed this means that $L\in C$, as wanted. Therefore the subset C of a normed vector space V is complete in V as every Cauchy sequence in C converges to a vector in C.