Question $1$:
Suppose that $(D,\le '), (E, \le '')$ are two directed sets where $D \cap E = \varnothing$. We impose an order $\le$ on $D \cup E$ thus: $x \le y \iff \cases{x, y \in D, x \le′ y \text{ or } \\ x, y \in E, x \le'' y \text{ or } \\ x \in D, y \in E}$
When will a net $(x_\gamma)_{\gamma \in D \cup E}$ converge in a space?
Question $2$:
Suppose that $(x_\gamma)_{\gamma \in D}$ is a convergent net in $(X, \tau)$, where $(D, \le)$ is a directed set. Take any $\gamma_0 \in D$, define $D_0 = \{\gamma \in D: \gamma \ge \gamma_0\}$ and order the elements of $D_0$ by the same order as in $D$. Show the net $(x_\gamma)_{\gamma \in D_0}$ also converges.
Answers:
Let $U$ be a neighborhood of limit $l$.
$1$:
Suppose,
exists $K \in D $ s.t. $k_i \ge K \implies x_{k_i} \in U$
and
exists $N \in E $ s.t. $n_i \ge N \implies x_{n_i} \in U$.
Then by definition, $K \le N \le n_i$ and so $x_{n_i} \in U$ and so exists $K \in D $ or exists $N \in E$ s.t. $n_i \ge K \implies x_{n_i} \in U$.
Thus we want $(x_\gamma)$ to converge on both $(D, \le')$ and $(E, \le '')$.
$2$:
Exists $\gamma_1 \in D $ s.t. $\gamma \ge \gamma_1 \implies x_\gamma \in U$. Fix this $\gamma_1$. Then by definition, $D_0 = \{\gamma \in D: \gamma \ge \gamma_1\}$ and so for all $\omega \in D_0$, we have $x_\omega \in U$ meaning $(x_\gamma)$ converges on $D_0$.
Do the answers above work? Thanks.
edit:
rewrite for $(2)$:
As $D_0$ inherits the order $\le$ from $D$, it follows $D_0$ is also directed and s0 $\gamma_0 \le \gamma_1 \le \gamma_2 \le \ldots \le \gamma_k \le \ldots$ holds for the elements in $D_0$. Now by definition of $D_0$ we have $\gamma_k \in D_0$ and by assumption we have $\gamma \ge \gamma_1 \implies x_\gamma \in U.$ Thus $\gamma (\ge \gamma_k) \in D_0 \implies \gamma \ge \gamma_1 \implies x_\gamma \in U.$ If $\gamma (< \gamma_k) \in D_0$, then the smallest possible value for $\gamma$ is $\gamma = \gamma_1$ by assumption(definition of convergence of $(x_\gamma))$. Otherwise, there is some $j$ with $2 \le j < k$ s.t. $\gamma_2 \le \gamma_j \le \gamma.$ In either case, $x_\gamma \in U.$
No, in 1. $D$ does not matter as $E$ is cofinal in $D \cup E$. In fact $(x_\gamma)_{\gamma \in D \cup E}$ converges to $p$ iff $(x_\gamma)_{\gamma \in E}$ converges to $p$, or more simply put a net $f: D \cup E \to X$ converges to $p$ iff $f\restriction_E$ does.
For 2. you need to explicitly invoke the directness of $D$ to find a $\gamma_2$ so that $\gamma_2 \ge \gamma_1$ and $\gamma_2 \ge \gamma_0$ and use that $\gamma_2$ for $U$ (when $\gamma_1$ works for $U$ in the larger net).
So no, both your answers need corrections...