(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Suppose k>2,x,yin R^k, |x-y| = d > 0, and r > 0.

Prove if 2r > d, there are infinitely manyzin 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 onez.

- Suppose there existed one and only onez. Is there a contradiction? Or, can I findz', a linear combination ofzthat also works and then fromz'use the same rule to getz'', 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 ordin R^k such thatz'= cz+dalso 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?

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# Homework Help: Real Analysis (Rudin) exercise with inequalities

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