Generalizing open subsets to $\Bbb{R}$ Prove that if $X$ and $Y$ are open subsets of $\Bbb{R}$ then $X\times Y$ is an open subset of $\Bbb{R}^2$. State and prove a generalization to $\Bbb{R}^n$.
I figure the best approach is just to start with the generalization. 
So if $X_1,...,X_n$ are open subsets of $\Bbb{R}^n$ then $X_1\times\cdots\times X_n$ are open subsets of $\Bbb{R}^n$. Prove this statement. 
This is where I'm struggling to figure out what to prove exactly. Do I use open balls? Or perhaps the complement? I solved a similar problem involving proving all subsets were closed.

I'm going to give a big hint on doing it for $n=2$, and you can work on the generalization.
How do you show a subset $U$ of $\Bbb R^n$ is open? You should that given $x\in U$, there exists $r>0$ such that $B_{r}(x)$, the ball of radius $r$, is contained in $U$.
To this end, let $(x,y)\in X\times Y$. Then there exists $r_1>0$ such that $B_{r_1}(x)\subseteq X$ (why?). Similarly there exists $r_2>0$ such that $B_{r_2}(y)\subseteq Y$.
Now, we need an open ball in $\Bbb R^2$. Take a suitable choice of $r$ (really your only candidate is either the minimum or maximum of $\{r_1,r_2\}$), and try to show that $B_r((x,y))\subseteq X\times Y$. It may be easier to show that $B_r((x,y))\subseteq B_{r_1}(x)\times B_{r_2}(y)$.
