Sum of Closures is Subset of Closure of Sum in Topological Vector Space/Proof 2

Proof
Let $x \in A^-$ and $y \in B^-$.

From Point in Set Closure iff Limit of Moore-Smith Sequence, there exists directed sets $\struct {\Lambda_1, \preceq_1}$ and $\struct {\Lambda_2, \preceq_2}$ and Moore-Smith sequences $\sequence {x_\lambda}_{\lambda \mathop \in \Lambda_1}$ valued in $A$ and $\sequence {y_\lambda}_{\lambda \mathop \in \Lambda_2}$ valued in $B$ such that:
 * $\sequence {x_\lambda}_{\lambda \mathop \in \Lambda_1}$ converges to $x$
 * $\sequence {y_\lambda}_{\lambda \mathop \in \Lambda_2}$ converges to $y$.

Define a relation $\sqsubseteq$ on $\Lambda_1 \times \Lambda_2$ by:


 * $\tuple {\lambda_1, \lambda_2} \sqsubseteq \tuple {\mu_1, \mu_2}$

for $\tuple {\lambda_1, \lambda_2}, \tuple {\mu_1, \mu_2} \in \Lambda_1 \times \Lambda_2$ :


 * $\lambda_1 \preceq_1 \mu_1$ and $\lambda_2 \preceq_2 \mu_2$

From Product of Directed Sets is Directed Set, $\struct {\Lambda_1 \times \Lambda_2, \sqsubseteq}$ is a directed set.

Define:
 * $x'_{\tuple {\lambda, \mu} } = x_\lambda$

and:
 * $y'_{\tuple {\lambda, \mu} } = y_\mu$

for each $\tuple {\lambda, \mu} \in \Lambda_1 \times \Lambda_2$.

We show that $\langle {x'_{\tuple {\lambda, \mu} } \rangle}_{\tuple {\lambda, \mu} \mathop \in \Lambda_1 \times \Lambda_2}$ converges to $x$.

Fix $\mu_0 \in \Lambda_2$.

Let $U$ be an open neighborhood of $x$.

Since $\sequence {x_\lambda}_{\lambda \mathop \in \Lambda_1}$, there exists $\lambda_0 \in \Lambda$ such that for all $\lambda \in \Lambda$ with $\lambda \succeq \lambda_0$, we have $x_\lambda \in U$.

Then, if $\tuple {\lambda, \mu} \in \Lambda_1 \times \Lambda_2$ has $\tuple {\lambda, \mu} \sqsupseteq \tuple {\lambda_0, \mu_0}$, we in particular have $\lambda \succeq \lambda_0$ and so $x'_{\tuple {\lambda, \mu} } \in U$.

Since $U$ was arbitrary, $\sequence {x'_{\tuple {\lambda, \mu} } }_{\tuple {\lambda, \mu} \mathop \in \Lambda_1 \times \Lambda_2}$ converges to $x$.

Similarly, $\sequence {y'_{\tuple {\lambda, \mu} } }_{\tuple {\lambda, \mu} \mathop \in \Lambda_1 \times \Lambda_2}$ converges to $y$.

From Sum of Convergent Moore-Smith Sequences in Topological Vector Space is Convergent, $\sequence {x'_{\tuple {\lambda, \mu} } + y'_{\tuple {\lambda, \mu} } }_{\tuple {\lambda, \mu} \mathop \in \Lambda_1 \times \Lambda_2}$ converges to $x + y$.

Since $x'_{\tuple {\lambda, \mu} } \in A$ and $y'_{\tuple {\lambda, \mu} } \in B$ for all $\tuple {\lambda, \mu} \in \Lambda_1 \times \Lambda_2$, we have that $x'_{\tuple {\lambda, \mu} } + y'_{\tuple {\lambda, \mu} } \in A + B$ for each $\tuple {\lambda, \mu} \in \Lambda_1 \times \Lambda_2$.

So $x + y$ is the limit of a Moore-Smith sequence valued in $A + B$.

So by Point in Set Closure iff Limit of Moore-Smith Sequence, we have $x + y \in \paren {A + B}^-$.

So, we have $A^- + B^- \subseteq \paren {A + B}^-$.