Show that I^n (unit cube) and S^n (n > 0) are both path-connected by constructing a path.
The cube you can literally just do a piecewise linear map (go down the appropriate amount and over the appropriate amount), for $\displaystyle \mathbb{S}^n$ just paramaterize it with two charts and go from there. Try this and report back if you have any issues.
What do you mean by paramaterize it with two charts?
Sorry, if my question is silly
No, I'm sorry, it's not a silly question, and you really need more than two charts theoretically. But, even that's just stupid. If it's on a sphere, just think about a great circle which contains the two points and take either of the paths induced from that.