System Identification using LMS Adaptive Filter

I just have a question about using an least-mean-squares algorithim adaptive filter for system identification. Consider the following

I am told that as the error converges to a small value, the adaptive filter coefficients w[k] will indeed repersent the unknown system h[k]. Now, that doesn't make sense to me since the n[k] is being used in the error calculation.

Won't the adaptive filter coefficients w[k] now repersent the unknown system coefficients h[k] AND the noise n[k]?