You travel on Earth's surface south $n$ miles, then east $n$ miles, then north $n$ miles and find yourself back where you started, *without visiting any point more than twice*. What is the closest you could have been to the south pole when you started? Assume Earth is a sphere with radius $R>n$.

My initial thought for this question was that you could actually be on the south pole and make $n = 0$, but this seems like a silly answer. At the moment I am thinking that it is some infinitely small number like $0.000000....1$ because, in my mind, you could travel any value of $n$ miles around the south pole when you are this close to it and it wouldn't matter how far you have travelled south or north, because there are no restrictions to this.

I can't seem to get this into algebra or an actual answer though.