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13. Sample Size Determination: Two-Stage Procedures  575

                              Proof From the definition of N in (13.2.4), one can verify that



                           which is referred to as the basic inequality. Now, let us proceed with the
                           proof.
                              (i) Observe that one has













                           But, with                               for n ≥ m, we can write














                           Now,             and one can verify using L’Hôpital’s rule from (1.6.29)

                           that                      Hence, from (13.2.8) we conclude that
                           P µ,σ 2 {Y > a } = 0. Then, combining this with (13.2.7), we conclude that
                                     n
                           P µ,σ 2 {N = ∞} = 0 so that one has P µ,σ 2 {N < ∞} = 1.
                              (ii) We take expectations throughout the basic inequality in (13.2.6) and
                           use the fact that             We leave the details out as Exercise 13.2.1.
                              (iii) For any fixed x ∈ ℜ, let us write
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