Lemma: Support bound for fields with an initial lower bound

Let $F$ be an ambient field with $\mathrm{supp}F\subset \{t\ge 0\}$. Suppose a compact observation set $K$ is contained in $\{t\le b,|\mathbf{x}|\le R\}$, where $b,R\ge 0$. Then

and the intersection is compact.

Proof. For $y\in \mathrm{supp}F\cap J^{-}(K)$, there is $x\in K$ such that

Thus $|\mathbf{y}|\le R+b$.

The right side of (eq:pastbound) is compact. The support is closed. Also $J^{-}(K)$ is closed: if $y_j\to y$ and $y_j\in J^{-}(x_j)$ with $x_j\in K$, a subsequence of $x_j$ converges to $x\in K$, and the closed causal inequality passes to the limit. The intersection is therefore a closed subset of a compact set.