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: $V_n \subseteq V_{n - 1}$
We have:

Proof: $U_n \subseteq V_{n - 1} \circ U_{n - 1}$
We have:

Proof: $V_n \circ U_n \subseteq V_0$
Proceeds by induction on $n$.

For all $n \in \N_{> 0}$, let $\map P n$ be the proposition:
 * $V_n \circ U_n \subseteq V_0$

Basis for the Induction
$\map P 1$ is the case:

and $\map P 1$ is seen to hold.

This is our basis for the induction.

Induction Hypothesis
Now we need to show that, if $\map P k$ is true, where $k > 0$, then it logically follows that $\map P {k + 1}$ is true.

So this is our induction hypothesis:
 * $V_n \circ U_n \subseteq V_0$

Then we need to show:
 * $V_{n + 1} \circ U_{n + 1} \subseteq V_0$

Induction Step
This is our induction step:

So $\map P k \implies \map P {k + 1}$ and by the Principle of Mathematical Induction:
 * $(3):\quad\forall n \in \N_{> 0} : V_n \circ U_n \subseteq V_0$

Proof: $U_n \subseteq V_0$
We have: