Characterization of Paracompactness in T3 Space/Lemma 14

Theorem
Let $T = \struct{X, \tau}$ be a topological space.

Let $\tau_{X \times X}$ denote the product topology on the cartesian product $X \times X$.

Let $\sequence{V_n}_{n \in \N}$ be a sequence of neighborhoods of the diagonal $\Delta_X$ in $\struct {X \times X, \tau_{X \times X} }$:
 * $\forall n \in \N_{> 0} : V_n$ is symmetric as a relation on $X \times X$
 * $\forall n \in \N_{> 0}$ the composite relation $V_n \circ V_n$ is a subset of $V_{n - 1}$, that is, $V_n \circ V_n \subseteq V_{n - 1}$

For all $n \in \N_{> 0}$, let:
 * $U_n = V_n \circ V_{n - 1}, \circ \cdots \circ V_1$

Then:
 * $\forall n : U_n \subseteq V_0$

Proof
By definition:
 * $\forall n \in \N_{> 0} : U_{n + 1} = V_{n + 1} \circ U_n$