For positive constants $p$ and $q$, I have a set $C=\{(x_1,x_2)\in\mathbb{R}^2:x_1\leq p, x_2\leq q\}$, which I know is convex, but I'm struggling on how to show this formally.
Intuitively I know it is convex, because $C$ is just a rectangular region in the third quadrant of the Cartesian plane:
So I can take any combination of points in $x,y\in C$, and for $t\in[0,1]$, it will be true that $xt+(1-t)y \in C$. Again, I can see this visually, and a linear combination of two vectors in $C$ will still be in $C$ if each vector is multiplied by a positive constant.
I'm just new to proving things and the topic of convex sets, so I'm not sure how to get started on formally proving that $C$ is convex. Any hints on how to get started?
Let $x=(x_1,x_2),y=(y_1,y_2)\in C$. If $t\in (0,1)$, then
$$tx_1 + (1-t)y_1\leq tp+(1-t)p=p,$$ and $$tx_2 + (1-t)y_2\leq tq+(1-t)q=q.$$ Therefore $tx+(1-t)y\in C$.