How to formally show a triangular region in the plane is bounded?

Question

Consider the region $$ D=\{(x,y)\in \mathbb{R}^2|x\ge0, y\ge 0, y<1-x\}. $$ I know this is the region between the $x$ axis, the $y$ axis and $y=1-x$, where the $x$ and $y$ axis are included in $D$ but $y=1-x$ is not.
How can I formally show that $D$ is bounded? The only definition of boundedness I know is:
$D$ is bounded if there exists $M>0$ such that $D$ is contained in the ball of radius $M$.

Answer 1

Consider the ball of radius $M$ centered at $0$ $$ B_M=\{(x,y)\in \mathbb{R}^2: x^2+y^2 \leq M\}. $$ To see $D$ is bounded it is enough to prove that $D \subseteq B_M$ for some $M>0$. So consider $(x_0, y_0) \in D$, rewriting part of the definition of $D$ we must have $$x_0+y_0<1. $$ Since $x_0 \geq 0$ we must have $y_0 <1$ and since $y_0 \geq 0$ we must have $x_0<1$. Therefore $0 \leq x_0 <1$ and $0 \leq y_0 <1$. Thus $$x_0^2+y_0^2 < 2,$$ and so $(x_0,y_0) \in B_2$. Since $(x_0,y_0)$ was an arbitrary point in $D$ this means $D \subseteq B_2$, so $D$ is bounded.