Definition: Let $E \subseteq \Bbb R^n: U,V \subseteq \Bbb R^n $. $(U,V)$ is said to separate $E$ iff
$ U,V \neq \emptyset$
$ U,V =$ relatively open in $E$
$ U \cap V = \emptyset$
$ U \cup V = E$
If $E$ is not separated by any pair $(U,V)$ then $E$ is said to be connected.
Using this definition I need to show that a set $E$ is connected. $$E:= \left(\{0\} \times [-1,1]\right) \cup \left\{\left(x, \sin\frac 1x\right) : x \in (0,1)\right\}$$
This is on an exam review, so I'd really like to get this type of problem down for the upcoming exam. My attempt at a proof by contradiction was to see which condition of the iff statement I could break but thus far haven't yielded results. Any advice is appreciated!
Hint:
You can view $E$ as the closure of a path-connected (and hence connected) set.