# proof that a finite collection of sets will not suffice

Suppose that you cut $[0,1]$ into $A_{0},...,A_{n}$. Displacing the parts is simply translating them; you can suppose that you leave $A_{0}$ in place and translate all the others to the right. Let $\epsilon$ be the smallest translation length : if after translation the union contains $[0,1]$, necessarily $[0,\epsilon]\subset A_{0}$. A contradiction ensues.

Title proof that a finite collection of sets will not suffice ProofThatAFiniteCollectionOfSetsWillNotSuffice 2013-03-22 14:38:46 2013-03-22 14:38:46 rspuzio (6075) rspuzio (6075) 4 rspuzio (6075) Proof msc 28E99