A point interior to a rectangle does not lie on the rectangle's border

Let $z, w \in \mathbb{C}$ and consider the closed axis-parallel rectangle $R(z,w) = [z_{\mathrm{re}}, w_{\mathrm{re}}] \times [z_{\mathrm{im}}, w_{\mathrm{im}}]$ (as a subset of $\mathbb{C}$, with sides taken as unordered intervals), together with its border $\partial R(z,w)$, the union of its four edges. Suppose $p \in \mathbb{C}$ is a point such that $R(z,w)$ is a neighborhood of $p$, i.e. $R(z,w) \in \mathcal{N}(p)$. Then

In other words, if the rectangle contains an open set around $p$, then $p$ must lie in the interior and cannot be on any of the four boundary edges.

This lemma is part of the PNT+ rectangle toolkit underlying contour integration on rectangles: residue-type arguments require the pole to sit strictly inside the contour, and this statement converts the topological hypothesis "the rectangle is a neighborhood of $p$" into the combinatorial fact that $p$ avoids the boundary, so that the integrand is well-behaved on the contour itself.