Sum Rule for Convergent Nets

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Theorem

Let $\struct {G, +}$ be a commutative topological semigroup.

Let $\struct {\Lambda, \preceq}$ be a directed set.

Let $\family {g_\lambda}_{\lambda \mathop \in \Lambda}$ and $\family {h_\lambda}_{\lambda \mathop \in \Lambda}$ be an indexed family of elements in $G$.

Let $\family {g_\lambda}_{\lambda \mathop \in \Lambda}$ and $\family {h_\lambda}_{\lambda \mathop \in \Lambda}$ be convergent to the following limits:

$\ds \lim_{\lambda \mathop \in \Lambda} g_\lambda = a$

$\ds \lim_{\lambda \mathop \in \Lambda} h_\lambda = b$

Then:

the indexed family $\family {g_\lambda + h_\lambda}_{\lambda \mathop \in \Lambda}$ converges to the limit:

$\ds \lim_{\lambda \mathop \in \Lambda} g_\lambda + h_\lambda = a + b$

Proof

Let $U$ be an open neighborhood of $a + b$.

By definition of topological semigroup:

the binary operation $+ : G \times G \to G$ is continuous.

By definition of continuous mapping:

$\exists W, V$ open neighborhoods of $a$ and $b$ respectively:

$+ \sqbrk {W \times V} \subseteq U$

By definition of convergence:

$\exists \lambda_1 \in \Lambda : \forall \mu \in \Lambda : \lambda_1 \preceq \mu \implies g_\mu \in W$

and:

$\exists \lambda_2 \in \Lambda : \forall \mu \in \Lambda : \lambda_2 \preceq \mu \implies h_\mu \in V$

By definition of directed set:

$\exists \lambda \in \Lambda : \lambda_1, \lambda_2 \preceq \lambda$

By definition of directed set:

$\forall \mu \in \Lambda : \lambda \preceq \mu \implies \lambda_1, \lambda_2 \preceq \mu$

Hence $\forall \mu \in \Lambda$ such that $\lambda \preceq \mu$:

$g_\mu + h_\mu \in U$

Since $U$ was arbitrary, for all open neighborhoods $U$ of $a + b$:

$\exists \lambda \in \Lambda : \forall \mu \in \Lambda : \lambda \preceq \mu \leadsto g_\mu + h_\mu \in U$

By definition of limit of net:

$\ds \lim_{\lambda \in \Lambda} g_\lambda + h_\lambda = a + b$

$\blacksquare$