Condition (2.209) holding uniformly w.r.t. τ assures that Assumption 2.6.11 is verified. The numbers Mp, possibly depending on , are independent of Ts. Under Assumption 2.6.11, the Weierstrass M-test (Johnsonbaugh and Pfaffenberger 2002, Theorem 62.6) assures the uniform convergence of the series of functions of Ts
Therefore, the limit operation can be interchanged with the infinite sum
where, in the second equality the sufficient condition (2.209) for Assumption 2.6.11 is used.
From Lemma 2.6.12 we have
(not necessarily uniformly with respect to and m).
From Theorems 2.6.4 and 2.6.5 it follows that, for every fixed Ts, , and m,
Therefore, for ...