Corollary 1

  Suppose that a_n > 0 and b_n, $n = 1,2,3,\ldots$, exist such that as $n \to \infty$,

$$F_n (x) \Rightarrow F(x),$$

$$F_n (a_nx + b_n) \Rightarrow G(x),$$

where both F(x) and G(x) are proper frequency distributions. If a and b are any limit points of the sequences $\{ a_n\}$ and $\{ b_n\}$, then $0 < a < \infty$ and b is finite.

Proof: This result was shown during the proof of Theorem 1.