Let $(f_x)$ and $(g_x)$ be two nets on a directed set $X$.
Show if $f_x \rightarrow \eta \,\,\,$ and $g_x \rightarrow \zeta$ so $f_x+g_x \rightarrow \eta + \zeta$
For $(f_x)$ holds:
$$\forall \epsilon > 0 \,\,\, \exists x_1 \in X\,\,\, \forall x\in X:x \succ x_1 \Longrightarrow |f_x-\eta|<\epsilon$$
And for $(g_x)$ holds:
$$\forall \epsilon > 0 \,\,\, \exists x_2 \in X\,\,\, \forall x\in X:x \succ x_2 \Longrightarrow |g_x-\zeta|<\epsilon$$
From the 3rd axiom of directed sets we know:
$$\forall x,y \in X \,\,\,\exists z\in X: z\succ x \,\,\wedge\,\, z\succ y$$
This means
$$\exists x_3 \in X: x_3\succ x_1 \,\,\wedge\,\, x_3\succ x_2$$
So:
$$\forall \epsilon > 0 \,\,\, \exists x_3 \in X\,\,\, \forall x\in X:x \succ x_3 \Longrightarrow|f_x-\eta|<\epsilon$$
and
$$\forall \epsilon > 0 \,\,\, \exists x_3 \in X\,\,\, \forall x\in X:x \succ x_3 \Longrightarrow|g_x-\zeta|<\epsilon$$
$$|(f_x+g_x)-(\eta+\zeta)|=|(f_x-\eta)+(g_x-\zeta)|\le |f_x-\eta|+|g_x-\zeta|<2\epsilon$$
Lets call $\vartheta:=2\epsilon$
$$\Longrightarrow \forall \vartheta > 0 \,\,\, \exists x_3 \in X\,\,\, \forall x\in X:x \succ x_3 \Longrightarrow|(f_x+g_x)-(\eta+\zeta)|<\vartheta$$
Since the limit is distinct.
$$f_x+g_x \rightarrow \eta+\zeta$$
$\Box$
Would be great if someone could look over it, and give me feedback, if my work is correct, and if not, what I should improve! Thank you
It is almost correct but you shouldn't say call $\nu=2\epsilon$. You should start with $\nu >0$ and take $\epsilon =\frac { \nu} 2$ in your argument.