Is the projection of a cauchy sequence in a normed finite dimensional vector space along some subspace also cauchy?

Question

Let $V$ be a normed finite dimensional vector space. Let $S$ and $S'$ be two subspace such that $S \cap S'={0}$ and $V$ is the direct sum of $S$ and $S'$. We define the projection of a vector $x$ along $S$ as the vector $P(x)$ such that $x=P(x)+c$ such that $c$ lies in $S'$. Clearly, $P$ is a well defined linear transformation. So if we have a cauchy sequence $\{a_n\}$, is the sequence $\{P(a_n)\}$ also a cauchy sequence?

Answer 1

Surely, a projection being a linear continuous map, is Lipschitz, and so uniformly continuous. Now uniformly continuous maps take Cauchy sequences to Cauchy sequences.