Answer
This "proof" assumes the result to be proved. In an attempt to show that $\lfloor n/2\rfloor=(n-1)/2$, the author assumes the truth of the conjecture implicitly by writing the unjustified statement $\lfloor\frac{2k+1}{2}\rfloor=\frac{(2k+1)-1}{2}$. This is circular reasoning.
Work Step by Step
Note that, although the proposition is in fact true, the proof is still incorrect. To correct the proof, we can rewrite the faulty statement as $\lfloor\frac{2k+1}{2}\rfloor=\lfloor k+\frac{1}{2}\rfloor=k+\lfloor\frac{1}{2}\rfloor=k$, which is justified by Theorem 4.5.1.