1. The problem statement, all variables and given/known data Find the error in this proof and give an example in (ℝ,de) to illustrate where this proof breaks down. Proof that every totally bounded set in a metric space is bounded. The set S is totally bounded and can therefore be covered by finitely many balls of radius 1, say N balls of radius 1. Then S is a subset of any ball B(x,2N) provided X lies in S. Thus diam S≤4N so that S is bounded. I can't see the fault in the proof and therefore don't know where to start when looking for an example in (ℝ,de) that illustrates how the proof breaks down. Any suggestions?