the cartesian product of two bounded sets is bounded [closed]

Solution 1:

A subset of $\mathbb{R}^n$ is bounded iff it is contained in some ball $B(r)\subseteq\mathbb{R}^n$ centered at $0$ of radius $r>0$. Or equivalently when norm of every element is at most $r$.

So $X,Y\subseteq\mathbb{R}$ are bounded, meaning there is a ball $B(r_X)$ such that $X\subseteq B(r_X)$ and a ball $B(r_Y)$ such that $Y\subseteq B(r_Y)$.

Now consider $X\times Y$. If $(x,y)\in X\times Y$ then

$$\big\lVert (x,y)\big\rVert=\sqrt{x^2+y^2}\leq \sqrt{(|x|+|y|)^2}=\big||x|+|y|\big|\leq |x|+|y|< r_X+r_Y$$

In other words $(x,y)$ belongs the ball $B(r_X+r_Y)$. By the arbitrary choice of $(x,y)$ we conclude that $X\times Y$ is bounded.