Completeness of the bounded-variation space ($BV(\Omega )\text{ is Banach}$)

A Cauchy sequence $(u_n)$ in the BV space converges in $L^1$ to $u$. For fixed sufficiently large $n$, lower semicontinuity gives $|D(u_n-u)|(\Omega )\le {\mathrm{liminf}}_m|D(u_n-u_m)|(\Omega )$. The $L^1$ norm of the same difference converges, so the small full-norm Cauchy bound passes to $u_n-u$. The triangle inequality then puts $u$ in the BV space and proves convergence in its full norm. This argument turns completeness of $L^1$ and lower semicontinuity of variation into completeness of the combined norm.