I am using Axler's new book Measure, Integration & Real Analysis in my Real Variables class. The spirit of the proof below makes perfect sense but there's a minor detail that's bothering me -- shouldn't $\sum_{k=1}^{\infty} \ell(I_k)=4\epsilon$ not $2\epsilon$?
$\sum_{k=1}^{\infty} \ell(I_k)=\sum_{k=1}^{\infty} {\epsilon\over{2^k}}+{\epsilon\over{2^k}}=\sum_{k=1}^{\infty} {2\epsilon\over{2^k}}=2\epsilon\sum_{k=1}^{\infty} {1\over{2^k}}=2\epsilon({1\over{1-{1\over{2}}}})= 4\epsilon$
Am I going crazy?
