MIT OCW rocket question

The result found in problem 7.1 says that the velocity of the two stage rocket(v1 in the derivation in the link below) < velocity of a single stage rocket(v2). Am i misinterpreting the results since I thought that the purpose of a multistage rocket was to attain higher terminal velocities. If the derivation is wrong can you show me where because the mathematical operations and the logic behind it look fine to me.

http://ocw.mit.edu/courses/physics/8...ments/sol7.pdf