Textbook Problem

Complete the proof of case 2 of the strong induction argument in Example 11.5.5. In other words, show that if k is an odd integer and $w_i=\lfloor {\log }_{2}i\rfloor +1$ for every integer i with $1\le i\le k$, then $w_{k+1}=\lfloor {\log }_{2}k+1\rfloor +1$ .