any help will be appreciated for this problem. I have an idea what the solution should be, but I'm unsure if I have enough justification.

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Letp= (0,1) and q = (0,-1). LetXbe the subset of $\displaystyle R^2$ that is formed by intersecting all the closed Euclidean balls that containpandq. What isX?

Solution (so far):

We can immediately exclude closed Euclidean balls that do not have p and q as a boundary point, since there exist Euclidean balls contained in them, and that if X is contained in Y, the intersection is X.

With some basic geometry, I've shown that there are three balls that have p and q as boundary points: the unit ball centered at the origin, and balls of radius $\displaystyle sqrt(2)$ centered at (-1,0) and (1,0).

Now, I want to say that the intersection between the last two balls mentioned is X, but I cannot convince myself that is the case. Am I doing something wrong?