It is well-known that each finite-dimensional subspace $F$ of a (real or complex) normed linear space $(V,\lVert - \rVert)$ is closed. The standard proof is this.
$\lVert - \rVert$ induces a norm $\lVert - \rVert_F$ on $F$. Since all norms on $F$ are equivalent, $(F,\lVert - \rVert_F)$ is complete so that $F$ is closed in $(V,\lVert - \rVert)$.
Let us now consider a finite-dimensional $V$ and a codimension $1$ subspace $F$. I tried to find a "direct proof" that $F$ is closed in $V$, but all my attempts were in vain. By a direct proof I mean one based on elementary properties of norms (essentially the triangle inequality).
Question: Does there exist a proof not using the "heavy gun" that all norms on finite-dimensional linear spaces are equivalent?