1. The problem statement, all variables and given/known data Suppose k>2, x, y in R^k, |x-y| = d > 0, and r > 0. Prove if 2r > d, there are infinitely many z in R^k such that |z-x| = |z-y| = r (In Principles of Mathematical Analysis, it is problem 16(a) on page 23.) 2. Relevant equations |ax| = |a||x| |x-z| < or = |x-y| + |y-z| |x+y| < or = |x| + |y| 3. The attempt at a solution I'm not quite sure how to tackle this proof. Here's the general outline I have: - Show that there exists at least one z. - Suppose there existed one and only one z. Is there a contradiction? Or, can I find z', a linear combination of z that also works and then from z' use the same rule to get z'', ad infinitum? As you can see, I don't really know how to tackle "show there are infinitely many solutions" proofs. One thought I had was, well, suppose z satisfies the necessary conditions. Can I show that there exists c in R or d in R^k such that z' = cz + d also satisfies the necessary conditions? But all I can get is |z' - y| = |z - y + d| < or = |z-y| + |d| And |z' - y| has to = |z-y| = r, but that doesn't tell me anything. Can someone give a hint or two to point me in the right direction? How should I tackle this problem?