This is a continuation of the previous post available here. In the previous post, we develop the ingredients required for the vague convergence proof. Let us now return to the random matrix scenario.
Proof of Theorem 4: be a sequence of GUE matrices. Let be the eigenvalues of . Fix . Let us quickly evaluate the limit
Observe that by using Theorem 3, we have
where in the last line we use change of variable formula. Let us define
Note that are kernels since is a kernel. Let us also define . Then using proposition 1 (c) we have
Note that Proposition 4 implies that for every , uniformly over a ball of radius C in the complex plane. are entire functions. Hence and . Hence for we have
Hence by continuity lemma for fredholm determinants we have
where the measure is the lebesgue measure on the bounded interval . This completes the proof of Theorem 4