Show that the set of all $X\in\mathbb{R}^n$ such that $A\cdot X\geq c$, where $A\in\mathbb{R}^n\setminus\{0\}$ and $c\in\mathbb{R}$ is convex i.e. show any half plane is convex.
Pf:
Let $S:=\{X:A\in\mathbb{R}^n\setminus\{0\} \wedge A\cdot X\geq c\}$ and let $X_1,X_2\in S$. That is, assume $$A\cdot X_1\geq c\space\text{ and } A\cdot X_2\geq c$$
Now consider the line segment between $X_1,X_2$: $$(1-t)X_1+tX_2\space\text{ where $0\leq t\leq1$}$$ First observe $(1-t)X_1+tX_2\neq0$ for all $t$ since $X_1,X_2\neq0$ and $1-t$ and $t$ are never zero simultaneously. Now we show if $X(t)=(1-t)X_1+tX_2$, then $$A\cdot\left((1-t)X_1+tX_2\right)\geq c\implies X(t)\in S\space\text{for all $t$}$$
$$\iff (1-t)A\cdot X_1+tA\cdot X_2\geq(1-t)(c)+t(c)=c$$
Thus since the line segment joining any two points of $S$ is contained in $S$, we conclude $S$ is convex.
Is this an okay proof?