This paper covers the perfect indirect quantum measurement, specifically in the context of repeated measurement. The indirect measurement is useful as it allows information to be obtained from quantum systems without inflicting much disturbance on them. We restrict ourselves to cases with no evolution of the measured system between measurements and to perfect measurements, that is, measurements from which no outgoing information is missed and no extra information is added. In this case we can make use of the work by M. A. Nielsen (2005). It says that the expected amount of information following a perfect indirect measurement is larger than the information before the measurement. We make use of this result to show that the repeated indirect perfect measurement of a quantum state has two mutually exclusive outcomes. The first outcome is that the measured state becomes a pure state almost surely. The second is that the measurement eventually stops resulting in information being revealed. In the latter case, further measurements on the system result in the state switching through spaces with the same dimension, and thus it does not become a pure state. This paper builds on the work by Maassen and K¨ummerer from 2005, which already proved this, by expanding their proofs and adding additional theorems and proofs to create a more self-contained result. Further studies might look at the rate at which states become pure, and what might influence this rate.